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Mirrors > Home > ILE Home > Th. List > imaundir | GIF version |
Description: The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.) |
Ref | Expression |
---|---|
imaundir | ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 4522 | . . 3 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ran ((𝐴 ∪ 𝐵) ↾ 𝐶) | |
2 | resundir 4803 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) | |
3 | 2 | rneqi 4737 | . . 3 ⊢ ran ((𝐴 ∪ 𝐵) ↾ 𝐶) = ran ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) |
4 | rnun 4917 | . . 3 ⊢ ran ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶)) | |
5 | 1, 3, 4 | 3eqtri 2142 | . 2 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶)) |
6 | df-ima 4522 | . . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶) | |
7 | df-ima 4522 | . . 3 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶) | |
8 | 6, 7 | uneq12i 3198 | . 2 ⊢ ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶)) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶)) |
9 | 5, 8 | eqtr4i 2141 | 1 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 ∪ cun 3039 ran crn 4510 ↾ cres 4511 “ cima 4512 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 |
This theorem is referenced by: fvun1 5455 |
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