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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 4930 | . 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 98 = wceq 1242 Fn wfn 4840 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-fun 4847 df-fn 4848 |
This theorem is referenced by: fneq12d 4934 f1o00 5104 f1ompt 5263 fmpt2d 5270 f1ocnvd 5644 offval2 5668 ofrfval2 5669 caofinvl 5675 |
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