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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 4238 | . 2 ⊢ (𝐵 ∩ 𝐴) ⊆ 𝐴 | |
| 2 | fnssres 6691 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 ∩ 𝐴) ⊆ 𝐴) → (𝐹 ↾ (𝐵 ∩ 𝐴)) Fn (𝐵 ∩ 𝐴)) | |
| 3 | 1, 2 | mpan2 691 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐵 ∩ 𝐴)) Fn (𝐵 ∩ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∩ cin 3950 ⊆ wss 3951 ↾ cres 5687 Fn wfn 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-fun 6563 df-fn 6564 | 
| This theorem is referenced by: resfnfinfin 9377 resfifsupp 9437 hashresfn 14379 | 
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