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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 5344 | . 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 105 = wceq 1353 ⟶wf 5208 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-fun 5214 df-fn 5215 df-f 5216 |
This theorem is referenced by: feq12d 5351 fco2 5378 fssres2 5389 fresin 5390 fmpt3d 5668 fmptco 5678 fressnfv 5699 off 6089 caofinvl 6099 f2ndf 6221 eroprf 6622 pmresg 6670 fseq1p1m1 10080 mgmplusf 12677 mgmb1mgm1 12679 grpsubf 12838 lmbr 13380 blfps 13576 blf 13577 dvmptclx 13847 lgsfcl3 14089 |
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