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Mirrors > Home > MPE Home > Th. List > feq123d | 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) |
feq123d.3 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
feq123d | ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq12d.1 | . . 3 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | feq12d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 1, 2 | feq12d 6725 | . 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)) |
4 | feq123d.3 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
5 | 4 | feq3d 6724 | . 2 ⊢ (𝜑 → (𝐺:𝐵⟶𝐶 ↔ 𝐺:𝐵⟶𝐷)) |
6 | 3, 5 | bitrd 279 | 1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ⟶wf 6559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 |
This theorem is referenced by: feq123 6727 feq23d 6732 fprg 7175 csbwrdg 14579 funcestrcsetclem8 18203 funcsetcestrclem8 18218 funcsetcestrclem9 18219 evlfcl 18279 yonedalem3a 18331 yonedalem4c 18334 yonedalem3b 18336 yonedainv 18338 iscau 25324 isuhgr 29092 uhgreq12g 29097 isuhgrop 29102 uhgrun 29106 isupgr 29116 upgrop 29126 isumgr 29127 upgrun 29150 umgrun 29152 lfuhgr1v0e 29286 wlkp1 29714 sseqf 34374 ismfs 35534 isrngo 37884 gneispace2 44122 isubgruhgr 47792 funcringcsetcALTV2lem8 48141 funcringcsetclem8ALTV 48164 |
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