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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 6504 | . 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)) |
4 | feq123d.3 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
5 | 4 | feq3d 6503 | . 2 ⊢ (𝜑 → (𝐺:𝐵⟶𝐶 ↔ 𝐺:𝐵⟶𝐷)) |
6 | 3, 5 | bitrd 281 | 1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ⟶wf 6353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-fun 6359 df-fn 6360 df-f 6361 |
This theorem is referenced by: feq123 6506 feq23d 6511 fprg 6919 csbwrdg 13897 funcestrcsetclem8 17399 funcsetcestrclem8 17414 funcsetcestrclem9 17415 evlfcl 17474 yonedalem3a 17526 yonedalem4c 17529 yonedalem3b 17531 yonedainv 17533 iscau 23881 isuhgr 26847 uhgreq12g 26852 isuhgrop 26857 uhgrun 26861 isupgr 26871 upgrop 26881 isumgr 26882 upgrun 26905 umgrun 26907 lfuhgr1v0e 27038 wlkp1 27465 sseqf 31652 ismfs 32798 isrngo 35177 gneispace2 40489 funcringcsetcALTV2lem8 44321 funcringcsetclem8ALTV 44344 |
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