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how can we express the following correctly.

$$f:E\to F $$ $$E \subseteq C_1 \times C_2 \times C_3 \times C_4 $$

$$f|_A:A\to F$$ $$A \subseteq C_1 \times C_4 $$ here, I misused the notation of mathematical restriction. I am looking for some notation like $f|_A$

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