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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 4876 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) | |
2 | 1 | rneqd 4830 | . 2 ⊢ (𝐴 = 𝐵 → ran (𝐶 ↾ 𝐴) = ran (𝐶 ↾ 𝐵)) |
3 | df-ima 4614 | . 2 ⊢ (𝐶 “ 𝐴) = ran (𝐶 ↾ 𝐴) | |
4 | df-ima 4614 | . 2 ⊢ (𝐶 “ 𝐵) = ran (𝐶 ↾ 𝐵) | |
5 | 2, 3, 4 | 3eqtr4g 2222 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ran crn 4602 ↾ cres 4603 “ cima 4604 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2726 df-un 3118 df-in 3120 df-ss 3127 df-sn 3579 df-pr 3580 df-op 3582 df-br 3980 df-opab 4041 df-xp 4607 df-cnv 4609 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 |
This theorem is referenced by: imaeq2i 4941 imaeq2d 4943 ssimaex 5544 ssimaexg 5545 isoselem 5785 f1opw2 6041 fopwdom 6796 ssenen 6811 fiintim 6888 fidcenumlemrk 6913 fidcenumlemr 6914 sbthlem2 6917 isbth 6926 ennnfonelemp1 12333 ennnfonelemnn0 12349 ctinfomlemom 12354 ctinfom 12355 tgcn 12806 iscnp4 12816 cnpnei 12817 cnima 12818 cnconst2 12831 cnrest2 12834 cnptoprest 12837 txcnp 12869 txcnmpt 12871 metcnp3 13109 |
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