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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 2643 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ ∃𝑧∃𝑥 ∈ 𝐵 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) | |
2 | vex 2623 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 2 | elima3 4794 | . . . . 5 ⊢ (𝑦 ∈ (𝐴 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
4 | 3 | rexbii 2386 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 “ 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
5 | eliun 3740 | . . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑧 ∈ 𝐶) | |
6 | 5 | anbi1i 447 | . . . . . 6 ⊢ ((𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ (∃𝑥 ∈ 𝐵 𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
7 | r19.41v 2524 | . . . . . 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 1542 | . . . 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 4794 | . . 3 ⊢ (𝑦 ∈ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ ∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
12 | eliun 3740 | . . 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 2086 | 1 ⊢ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶) = ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1290 ∃wex 1427 ∈ wcel 1439 ∃wrex 2361 〈cop 3453 ∪ ciun 3736 “ cima 4455 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-iun 3738 df-br 3852 df-opab 3906 df-xp 4458 df-cnv 4460 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 |
This theorem is referenced by: imauni 5554 uniqs 6364 |
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