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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 6723 | . 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)) | 
| 4 | feq123d.3 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
| 5 | 4 | feq3d 6722 | . 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 1539 ⟶wf 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-fun 6562 df-fn 6563 df-f 6564 | 
| This theorem is referenced by: feq123 6725 feq23d 6730 fprg 7174 csbwrdg 14583 funcestrcsetclem8 18193 funcsetcestrclem8 18208 funcsetcestrclem9 18209 evlfcl 18268 yonedalem3a 18320 yonedalem4c 18323 yonedalem3b 18325 yonedainv 18327 iscau 25311 isuhgr 29078 uhgreq12g 29083 isuhgrop 29088 uhgrun 29092 isupgr 29102 upgrop 29112 isumgr 29113 upgrun 29136 umgrun 29138 lfuhgr1v0e 29272 wlkp1 29700 sseqf 34395 ismfs 35555 isrngo 37905 gneispace2 44150 isubgruhgr 47859 funcringcsetcALTV2lem8 48218 funcringcsetclem8ALTV 48241 | 
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