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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 5384 | . 2 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) |
3 | imaeq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | imaeq2d 5385 | . 2 ⊢ (𝜑 → (𝐵 “ 𝐶) = (𝐵 “ 𝐷)) |
5 | 2, 4 | eqtrd 2644 | 1 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 “ cima 5041 |
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-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: csbima12 5402 predeq123 5598 vdwpc 15522 dmdprd 18220 isunit 18480 qtopval 21308 limciun 23464 ig1pval 23736 ispth 26098 qqhval 29346 eulerpartgbij 29761 orvcval 29846 ballotlemrval 29906 ballotlemrinv0 29921 ballotlemrinv 29922 mthmval 30726 bj-projeq 32173 itg2addnclem2 32632 islmodfg 36657 heeq12 37090 isPth 40929 |
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