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Mirrors > Home > ILE Home > Th. List > fnssres | GIF version |
Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.) |
Ref | Expression |
---|---|
fnssres | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnssresb 5308 | . 2 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ 𝐵 ⊆ 𝐴)) | |
2 | 1 | biimpar 295 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ⊆ wss 3121 ↾ cres 4611 Fn wfn 5191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-res 4621 df-fun 5198 df-fn 5199 |
This theorem is referenced by: fnresin1 5310 fnresin2 5311 fssres 5371 fvreseq 5597 fnreseql 5604 ffvresb 5657 fnressn 5680 ofres 6073 tfrlem1 6285 frecrdg 6385 resixp 6709 resfnfinfinss 6915 suplocexprlemell 7668 seq3feq2 10419 reeff1 11656 upxp 13031 uptx 13033 cnmpt1st 13047 cnmpt2nd 13048 ioocosf1o 13534 |
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