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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 6076 | . 2 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) | 
| 3 | imaeq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
| 4 | 3 | imaeq2d 6077 | . 2 ⊢ (𝜑 → (𝐵 “ 𝐶) = (𝐵 “ 𝐷)) | 
| 5 | 2, 4 | eqtrd 2776 | 1 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 “ cima 5687 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 | 
| This theorem is referenced by: csbima12 6096 predeq123 6321 vdwpc 17019 dmdprd 20019 isunit 20374 qtopval 23704 limciun 25930 ig1pval 26216 ispth 29742 irngval 33736 qqhval 33974 eulerpartgbij 34375 orvcval 34461 ballotlemrval 34521 ballotlemrinv0 34536 ballotlemrinv 34537 mthmval 35581 bj-projeq 36994 itg2addnclem2 37680 islmodfg 43086 heeq12 43794 isgrim 47873 | 
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