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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 4770 | . 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 1289 “ cima 4441 |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-sn 3452 df-pr 3453 df-op 3455 df-br 3846 df-opab 3900 df-xp 4444 df-cnv 4446 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 |
This theorem is referenced by: imaeq12d 4775 nfimad 4783 elimasng 4800 ressn 4971 foima 5238 f1imacnv 5270 fvco2 5373 fsn2 5471 resfunexg 5518 funfvima3 5528 funiunfvdm 5542 isoselem 5599 fnexALT 5884 eceq1 6327 uniqs2 6352 ecinxp 6367 mapsn 6447 phplem4 6571 phplem4dom 6578 phplem4on 6583 sbthlem2 6667 isbth 6676 resunimafz0 10236 |
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