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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 6642 | . 2 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) |
3 | fneq12d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | fneq2d 6643 | . 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 205 = wceq 1540 Fn wfn 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-fun 6545 df-fn 6546 |
This theorem is referenced by: fneq12 6645 seqfn 13983 sscres 17775 reschomf 17784 funcres 17851 psrvscafval 21729 ressprdsds 24098 rrxmfval 25155 ex-fpar 29983 sseqfn 33688 tfsconcatfn 42391 funcoressn 46051 |
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