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Mirrors > Home > ILE Home > Th. List > imaeq2d | GIF version |
Description: Equality theorem for image. (Contributed by FL, 15-Dec-2006.) |
Ref | Expression |
---|---|
imaeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
imaeq2d | ⊢ (𝜑 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imaeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | imaeq2 4684 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 “ cima 4366 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-cnv 4371 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 |
This theorem is referenced by: imaeq12d 4689 nfimad 4697 elimasng 4713 ressn 4878 foima 5131 f1imacnv 5163 fvco2 5263 fsn2 5358 resfunexg 5403 funfvima3 5413 funiunfvdm 5423 isoselem 5479 fnexALT 5760 eceq1 6164 uniqs2 6189 ecinxp 6204 phplem4 6341 phplem4dom 6348 phplem4on 6353 |
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