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Mirrors > Home > MPE Home > Th. List > imaeq12d | Structured version Visualization version GIF version |
Description: Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.) |
Ref | Expression |
---|---|
imaeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
imaeq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
imaeq12d | ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imaeq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | imaeq1d 5928 | . 2 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) |
3 | imaeq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | imaeq2d 5929 | . 2 ⊢ (𝜑 → (𝐵 “ 𝐶) = (𝐵 “ 𝐷)) |
5 | 2, 4 | eqtrd 2856 | 1 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 “ cima 5558 |
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 2793 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 |
This theorem is referenced by: csbima12 5947 predeq123 6149 vdwpc 16316 dmdprd 19120 isunit 19407 qtopval 22303 limciun 24492 ig1pval 24766 ispth 27504 qqhval 31215 eulerpartgbij 31630 orvcval 31715 ballotlemrval 31775 ballotlemrinv0 31790 ballotlemrinv 31791 mthmval 32822 bj-projeq 34307 itg2addnclem2 34959 islmodfg 39689 heeq12 40142 |
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