Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 6030 | . 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵)) |
4 | 1, 3 | mpbid 222 | 1 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ⟶wf 5884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-fun 5890 df-fn 5891 df-f 5892 |
This theorem is referenced by: cncficcgt0 40101 itgsubsticclem 40191 itgsbtaddcnst 40198 fourierdlem103 40426 fourierdlem104 40427 fourierdlem113 40436 ismeannd 40684 hoidmv1le 40808 |
Copyright terms: Public domain | W3C validator |