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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 5367 | . 2 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ 𝐵 ⊆ 𝐴)) | |
2 | 1 | biimpar 297 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ⊆ wss 3154 ↾ cres 4662 Fn wfn 5250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-br 4031 df-opab 4092 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-res 4672 df-fun 5257 df-fn 5258 |
This theorem is referenced by: fnresin1 5369 fnresin2 5370 fssres 5430 fvreseq 5662 fnreseql 5669 ffvresb 5722 fnressn 5745 ofres 6147 tfrlem1 6363 frecrdg 6463 resixp 6789 resfnfinfinss 7000 suplocexprlemell 7775 seq3feq2 10550 seqf1oglem2 10594 reeff1 11846 rngmgpf 13436 mgpf 13510 upxp 14451 uptx 14453 cnmpt1st 14467 cnmpt2nd 14468 ioocosf1o 15030 |
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