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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 5946 | . 2 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)) |
4 | feq123d.3 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
5 | 4 | feq3d 5945 | . 2 ⊢ (𝜑 → (𝐺:𝐵⟶𝐶 ↔ 𝐺:𝐵⟶𝐷)) |
6 | 3, 5 | bitrd 267 | 1 ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ⟶wf 5800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 |
This theorem is referenced by: feq123 5948 feq23d 5953 fprg 6327 csbwrdg 13189 funcestrcsetclem8 16610 funcsetcestrclem8 16625 funcsetcestrclem9 16626 evlfcl 16685 yonedalem3a 16737 yonedalem4c 16740 yonedalem3b 16742 yonedainv 16744 iscau 22882 isuhgr 25726 uhgreq12g 25731 isuhgrop 25736 uhgrun 25740 isupgr 25751 isumgr 25761 upgrun 25784 umgrun 25786 isuhgra 25827 uhgraeq12d 25836 constr3trllem3 26180 sseqf 29781 ismfs 30700 isrngo 32866 gneispace2 37450 uhgruhgra 40375 uhgrauhgr 40376 lfuhgr1v0e 40480 1wlkp1 40890 funcringcsetcALTV2lem8 41835 funcringcsetclem8ALTV 41858 |
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