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Mirrors > Home > ILE Home > Th. List > imaeq2 | GIF version |
Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.) |
Ref | Expression |
---|---|
imaeq2 | ⊢ (𝐴 = 𝐵 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reseq2 4822 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) | |
2 | 1 | rneqd 4776 | . 2 ⊢ (𝐴 = 𝐵 → ran (𝐶 ↾ 𝐴) = ran (𝐶 ↾ 𝐵)) |
3 | df-ima 4560 | . 2 ⊢ (𝐶 “ 𝐴) = ran (𝐶 ↾ 𝐴) | |
4 | df-ima 4560 | . 2 ⊢ (𝐶 “ 𝐵) = ran (𝐶 ↾ 𝐵) | |
5 | 2, 3, 4 | 3eqtr4g 2198 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵)) |
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-xp 4553 df-cnv 4555 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 |
This theorem is referenced by: imaeq2i 4887 imaeq2d 4889 ssimaex 5490 ssimaexg 5491 isoselem 5729 f1opw2 5984 fopwdom 6738 ssenen 6753 fiintim 6825 fidcenumlemrk 6850 fidcenumlemr 6851 sbthlem2 6854 isbth 6863 ennnfonelemp1 11955 ennnfonelemnn0 11971 ctinfomlemom 11976 ctinfom 11977 tgcn 12416 iscnp4 12426 cnpnei 12427 cnima 12428 cnconst2 12441 cnrest2 12444 cnptoprest 12447 txcnp 12479 txcnmpt 12481 metcnp3 12719 |
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