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Mirrors > Home > ILE Home > Th. List > feq1d | GIF version |
Description: Equality deduction for functions. (Contributed by NM, 19-Feb-2008.) |
Ref | Expression |
---|---|
feq1d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
Ref | Expression |
---|---|
feq1d | ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq1d.1 | . 2 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | feq1 5255 | . 2 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ⟶wf 5119 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-fun 5125 df-fn 5126 df-f 5127 |
This theorem is referenced by: feq12d 5262 fco2 5289 fssres2 5300 fresin 5301 fmpt3d 5576 fmptco 5586 fressnfv 5607 off 5994 caofinvl 6004 f2ndf 6123 eroprf 6522 pmresg 6570 fseq1p1m1 9874 lmbr 12382 blfps 12578 blf 12579 dvmptclx 12849 |
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