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Mirrors > Home > MPE Home > Th. List > fneq12d | Structured version Visualization version GIF version |
Description: Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.) |
Ref | Expression |
---|---|
fneq12d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
fneq12d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
fneq12d | ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fneq12d.1 | . . 3 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | 1 | fneq1d 6416 | . 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) |
3 | fneq12d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | fneq2d 6417 | . 2 ⊢ (𝜑 → (𝐺 Fn 𝐴 ↔ 𝐺 Fn 𝐵)) |
5 | 2, 4 | bitrd 282 | 1 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 Fn wfn 6319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-fun 6326 df-fn 6327 |
This theorem is referenced by: fneq12 6419 seqfn 13376 sscres 17085 reschomf 17093 funcres 17158 psrvscafval 20628 ressprdsds 22978 rrxmfval 24010 ex-fpar 28247 sseqfn 31758 funcoressn 43634 |
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