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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 4228 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | fnssres 6673 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵)) | |
3 | 1, 2 | mpan2 688 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3947 ⊆ wss 3948 ↾ cres 5678 Fn wfn 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-res 5688 df-fun 6545 df-fn 6546 |
This theorem is referenced by: frrlem4 8280 wfrlem4OLD 8318 fnresin 32285 |
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