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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 6483 | . 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵)) |
4 | 1, 3 | mpbid 235 | 1 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ⟶wf 6329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3399 df-un 3846 df-in 3848 df-ss 3858 df-sn 4514 df-pr 4516 df-op 4520 df-br 5028 df-opab 5090 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-fun 6335 df-fn 6336 df-f 6337 |
This theorem is referenced by: cncficcgt0 42955 itgsubsticclem 43042 itgsbtaddcnst 43049 fourierdlem103 43276 fourierdlem104 43277 fourierdlem113 43286 ismeannd 43531 hoidmv1le 43658 |
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