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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 6661 | . 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) |
| 3 | fneq12d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 4 | 3 | fneq2d 6662 | . 2 ⊢ (𝜑 → (𝐺 Fn 𝐴 ↔ 𝐺 Fn 𝐵)) |
| 5 | 2, 4 | bitrd 279 | 1 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 Fn wfn 6556 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-fun 6563 df-fn 6564 |
| This theorem is referenced by: fneq12 6664 seqfn 14054 sscres 17867 reschomf 17875 funcres 17941 psrvscafval 21968 ressprdsds 24381 rrxmfval 25440 ex-fpar 30481 sseqfn 34392 tfsconcatfn 43351 funcoressn 47054 |
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