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Mirrors > Home > ILE Home > Th. List > rniun | GIF version |
Description: The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.) |
Ref | Expression |
---|---|
rniun | ⊢ ran ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ran 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom4 2683 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦〈𝑦, 𝑧〉 ∈ 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
2 | vex 2663 | . . . . . 6 ⊢ 𝑧 ∈ V | |
3 | 2 | elrn2 4751 | . . . . 5 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑦〈𝑦, 𝑧〉 ∈ 𝐵) |
4 | 3 | rexbii 2419 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦〈𝑦, 𝑧〉 ∈ 𝐵) |
5 | eliun 3787 | . . . . 5 ⊢ (〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
6 | 5 | exbii 1569 | . . . 4 ⊢ (∃𝑦〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) |
7 | 1, 4, 6 | 3bitr4ri 212 | . . 3 ⊢ (∃𝑦〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵) |
8 | 2 | elrn2 4751 | . . 3 ⊢ (𝑧 ∈ ran ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) |
9 | eliun 3787 | . . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ran 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵) | |
10 | 7, 8, 9 | 3bitr4i 211 | . 2 ⊢ (𝑧 ∈ ran ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ran 𝐵) |
11 | 10 | eqriv 2114 | 1 ⊢ ran ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ran 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 ∃wex 1453 ∈ wcel 1465 ∃wrex 2394 〈cop 3500 ∪ ciun 3783 ran crn 4510 |
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-cnv 4517 df-dm 4519 df-rn 4520 |
This theorem is referenced by: rnuni 4920 fun11iun 5356 ennnfonelemrn 11859 |
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