Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5465 | . 2 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) |
3 | imaeq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | imaeq2d 5466 | . 2 ⊢ (𝜑 → (𝐵 “ 𝐶) = (𝐵 “ 𝐷)) |
5 | 2, 4 | eqtrd 2656 | 1 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 “ cima 5117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 |
This theorem is referenced by: csbima12 5483 predeq123 5681 vdwpc 15684 dmdprd 18397 isunit 18657 qtopval 21498 limciun 23658 ig1pval 23932 ispth 26619 qqhval 30018 eulerpartgbij 30434 orvcval 30519 ballotlemrval 30579 ballotlemrinv0 30594 ballotlemrinv 30595 mthmval 31472 bj-projeq 32980 itg2addnclem2 33462 islmodfg 37639 heeq12 38070 |
Copyright terms: Public domain | W3C validator |