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Mirrors > Home > MPE Home > Th. List > feq12d | Structured version Visualization version GIF version |
Description: Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
feq12d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
feq12d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
feq12d | ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq12d.1 | . . 3 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | 1 | feq1d 6713 | . 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐴⟶𝐶)) |
3 | feq12d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | feq2d 6714 | . 2 ⊢ (𝜑 → (𝐺:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)) |
5 | 2, 4 | bitrd 278 | 1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ⟶wf 6550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-br 5154 df-opab 5216 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-fun 6556 df-fn 6557 df-f 6558 |
This theorem is referenced by: feq123d 6717 fprg 7169 smoeq 8380 oif 9573 1fv 13674 catcisolem 18132 hofcl 18284 dmdprd 19998 dpjf 20057 pjf2 21712 mat1dimmul 22469 lmbr2 23254 lmff 23296 dfac14 23613 lmmbr2 25278 lmcau 25332 perfdvf 25923 dvnfre 25975 dvle 26031 dvfsumle 26045 dvfsumleOLD 26046 dvfsumge 26047 dvmptrecl 26049 uhgr0e 29007 uhgrstrrepe 29014 incistruhgr 29015 upgr1e 29049 1hevtxdg1 29443 umgr2v2e 29462 iswlk 29547 0wlkons1 30054 resf1o 32644 ismeas 34032 omsmeas 34157 breprexplema 34476 satfun 35239 mbfresfi 37367 sdclem1 37444 dfac21 42727 fnlimfvre 45295 climrescn 45369 fourierdlem74 45801 fourierdlem103 45830 fourierdlem104 45831 sge0iunmpt 46039 ismea 46072 isome 46115 sssmf 46359 smflimlem3 46394 smflimlem4 46395 isupwlk 47513 |
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