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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 4885 | . 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 1332 “ cima 4550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 |
This theorem is referenced by: imaeq12d 4890 nfimad 4898 elimasng 4915 ressn 5087 foima 5358 f1imacnv 5392 fvco2 5498 fsn2 5602 resfunexg 5649 funfvima3 5659 funiunfvdm 5672 isoselem 5729 fnexALT 6019 eceq1 6472 uniqs2 6497 ecinxp 6512 mapsn 6592 phplem4 6757 phplem4dom 6764 phplem4on 6769 sbthlem2 6854 isbth 6863 resunimafz0 10606 ennnfonelemg 11952 ennnfonelemhf1o 11962 ennnfonelemex 11963 ennnfonelemrn 11968 cnntr 12433 cnptopresti 12446 cnptoprest 12447 |
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