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Mirrors > Home > ILE Home > Th. List > imaiun | GIF version |
Description: The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
imaiun | ⊢ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶) = ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom4 2683 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ ∃𝑧∃𝑥 ∈ 𝐵 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) | |
2 | vex 2663 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 2 | elima3 4858 | . . . . 5 ⊢ (𝑦 ∈ (𝐴 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
4 | 3 | rexbii 2419 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 “ 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
5 | eliun 3787 | . . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑧 ∈ 𝐶) | |
6 | 5 | anbi1i 453 | . . . . . 6 ⊢ ((𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ (∃𝑥 ∈ 𝐵 𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
7 | r19.41v 2564 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝐵 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ (∃𝑥 ∈ 𝐵 𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) | |
8 | 6, 7 | bitr4i 186 | . . . . 5 ⊢ ((𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ ∃𝑥 ∈ 𝐵 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
9 | 8 | exbii 1569 | . . . 4 ⊢ (∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ ∃𝑧∃𝑥 ∈ 𝐵 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
10 | 1, 4, 9 | 3bitr4ri 212 | . . 3 ⊢ (∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 “ 𝐶)) |
11 | 2 | elima3 4858 | . . 3 ⊢ (𝑦 ∈ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ ∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
12 | eliun 3787 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝐶) ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 “ 𝐶)) | |
13 | 10, 11, 12 | 3bitr4i 211 | . 2 ⊢ (𝑦 ∈ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝐶)) |
14 | 13 | eqriv 2114 | 1 ⊢ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶) = ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1316 ∃wex 1453 ∈ wcel 1465 ∃wrex 2394 〈cop 3500 ∪ ciun 3783 “ 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-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-iun 3785 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 |
This theorem is referenced by: imauni 5630 uniqs 6455 |
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