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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 5310 | . 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 4613 Fn wfn 5193 |
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 4107 ax-pow 4160 ax-pr 4194 |
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 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-res 4623 df-fun 5200 df-fn 5201 |
This theorem is referenced by: fnresin1 5312 fnresin2 5313 fssres 5373 fvreseq 5599 fnreseql 5606 ffvresb 5659 fnressn 5682 ofres 6075 tfrlem1 6287 frecrdg 6387 resixp 6711 resfnfinfinss 6917 suplocexprlemell 7675 seq3feq2 10426 reeff1 11663 upxp 13066 uptx 13068 cnmpt1st 13082 cnmpt2nd 13083 ioocosf1o 13569 |
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