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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 6079 | . 2 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) |
3 | imaeq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | imaeq2d 6080 | . 2 ⊢ (𝜑 → (𝐵 “ 𝐶) = (𝐵 “ 𝐷)) |
5 | 2, 4 | eqtrd 2775 | 1 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 “ cima 5692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 |
This theorem is referenced by: csbima12 6099 predeq123 6324 vdwpc 17014 dmdprd 20033 isunit 20390 qtopval 23719 limciun 25944 ig1pval 26230 ispth 29756 irngval 33700 qqhval 33935 eulerpartgbij 34354 orvcval 34439 ballotlemrval 34499 ballotlemrinv0 34514 ballotlemrinv 34515 mthmval 35560 bj-projeq 36975 itg2addnclem2 37659 islmodfg 43058 heeq12 43766 isgrim 47806 |
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