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Mirrors > Home > MPE Home > Th. List > fnresin1 | Structured version Visualization version GIF version |
Description: Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.) |
Ref | Expression |
---|---|
fnresin1 | ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 4119 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | fnssres 6459 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵)) | |
3 | 1, 2 | mpan2 691 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3842 ⊆ wss 3843 ↾ cres 5527 Fn wfn 6334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-res 5537 df-fun 6341 df-fn 6342 |
This theorem is referenced by: wfrlem4 7989 fnresin 30537 frrlem4 33448 |
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