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Mirrors > Home > ILE Home > Th. List > imaeq1 | GIF version |
Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.) |
Ref | Expression |
---|---|
imaeq1 | ⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reseq1 4821 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) | |
2 | 1 | rneqd 4776 | . 2 ⊢ (𝐴 = 𝐵 → ran (𝐴 ↾ 𝐶) = ran (𝐵 ↾ 𝐶)) |
3 | df-ima 4560 | . 2 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶) | |
4 | df-ima 4560 | . 2 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶) | |
5 | 2, 3, 4 | 3eqtr4g 2198 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ran crn 4548 ↾ cres 4549 “ 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-cnv 4555 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 |
This theorem is referenced by: imaeq1i 4886 imaeq1d 4888 eceq2 6474 iscnp 12407 |
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