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Mirrors > Home > ILE Home > Th. List > fneq1d | GIF version |
Description: Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
fneq1d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
Ref | Expression |
---|---|
fneq1d | ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fneq1d.1 | . 2 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | fneq1 5181 | . 2 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1316 Fn wfn 5088 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-fun 5095 df-fn 5096 |
This theorem is referenced by: fneq12d 5185 f1o00 5370 f1ompt 5539 fmpt2d 5550 f1ocnvd 5940 offval2 5965 ofrfval2 5966 caofinvl 5972 f1od2 6100 neif 12237 |
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