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Mirrors > Home > MPE Home > Th. List > funin | Structured version Visualization version GIF version |
Description: The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (Contributed by NM, 19-Mar-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
funin | ⊢ (Fun 𝐹 → Fun (𝐹 ∩ 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 3833 | . 2 ⊢ (𝐹 ∩ 𝐺) ⊆ 𝐹 | |
2 | funss 5907 | . 2 ⊢ ((𝐹 ∩ 𝐺) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹 ∩ 𝐺))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (Fun 𝐹 → Fun (𝐹 ∩ 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3573 ⊆ wss 3574 Fun wfun 5882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 df-ss 3588 df-br 4654 df-opab 4713 df-rel 5121 df-cnv 5122 df-co 5123 df-fun 5890 |
This theorem is referenced by: (None) |
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