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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 4947 | . 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 1348 “ cima 4612 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-xp 4615 df-cnv 4617 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 |
This theorem is referenced by: imaeq12d 4952 nfimad 4960 elimasng 4977 ressn 5149 foima 5423 f1imacnv 5457 fvco2 5563 fsn2 5667 resfunexg 5714 funfvima3 5726 funiunfvdm 5739 isoselem 5796 fnexALT 6087 eceq1 6544 uniqs2 6569 ecinxp 6584 mapsn 6664 phplem4 6829 phplem4dom 6836 phplem4on 6841 sbthlem2 6931 isbth 6940 resunimafz0 10753 ennnfonelemg 12345 ennnfonelemhf1o 12355 ennnfonelemex 12356 ennnfonelemrn 12361 cnntr 12978 cnptopresti 12991 cnptoprest 12992 |
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