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Mirrors > Home > MPE Home > Th. List > Mathboxes > feq1dd | Structured version Visualization version GIF version |
Description: Equality deduction for functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
feq1dd.eq | ⊢ (𝜑 → 𝐹 = 𝐺) |
feq1dd.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
Ref | Expression |
---|---|
feq1dd | ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq1dd.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | feq1dd.eq | . . 3 ⊢ (𝜑 → 𝐹 = 𝐺) | |
3 | 2 | feq1d 6585 | . 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵)) |
4 | 1, 3 | mpbid 231 | 1 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ⟶wf 6429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-fun 6435 df-fn 6436 df-f 6437 |
This theorem is referenced by: cncficcgt0 43429 itgsubsticclem 43516 itgsbtaddcnst 43523 fourierdlem103 43750 fourierdlem104 43751 fourierdlem113 43760 ismeannd 44005 hoidmv1le 44132 |
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