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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 6440 | . 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) |
3 | fneq12d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | fneq2d 6441 | . 2 ⊢ (𝜑 → (𝐺 Fn 𝐴 ↔ 𝐺 Fn 𝐵)) |
5 | 2, 4 | bitrd 281 | 1 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 Fn wfn 6344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-fun 6351 df-fn 6352 |
This theorem is referenced by: fneq12 6443 seqfn 13375 sscres 17087 reschomf 17095 funcres 17160 psrvscafval 20164 ressprdsds 22975 rrxmfval 24003 ex-fpar 28235 sseqfn 31643 funcoressn 43271 |
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