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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 6218 | . 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) |
3 | fneq12d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | fneq2d 6219 | . 2 ⊢ (𝜑 → (𝐺 Fn 𝐴 ↔ 𝐺 Fn 𝐵)) |
5 | 2, 4 | bitrd 271 | 1 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1656 Fn wfn 6122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-fun 6129 df-fn 6130 |
This theorem is referenced by: fneq12 6221 seqfn 13114 sscres 16842 reschomf 16850 funcres 16915 psrvscafval 19758 ressprdsds 22553 rrxmfval 23581 sseqfn 30994 funcoressn 41971 |
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