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Mirrors > Home > MPE Home > Th. List > fnresin2 | Structured version Visualization version GIF version |
Description: Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.) |
Ref | Expression |
---|---|
fnresin2 | ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐵 ∩ 𝐴)) Fn (𝐵 ∩ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss2 4169 | . 2 ⊢ (𝐵 ∩ 𝐴) ⊆ 𝐴 | |
2 | fnssres 6553 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 ∩ 𝐴) ⊆ 𝐴) → (𝐹 ↾ (𝐵 ∩ 𝐴)) Fn (𝐵 ∩ 𝐴)) | |
3 | 1, 2 | mpan2 688 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐵 ∩ 𝐴)) Fn (𝐵 ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3891 ⊆ wss 3892 ↾ cres 5592 Fn wfn 6427 |
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 2015 ax-8 2112 ax-9 2120 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-br 5080 df-opab 5142 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-res 5602 df-fun 6434 df-fn 6435 |
This theorem is referenced by: resfnfinfin 9077 resfifsupp 9134 hashresfn 14052 |
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