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Mirrors > Home > ILE Home > Th. List > iunopab | GIF version |
Description: Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.) |
Ref | Expression |
---|---|
iunopab | ⊢ ∪ 𝑧 ∈ 𝐴 {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elopab 4150 | . . . . 5 ⊢ (𝑤 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
2 | 1 | rexbii 2419 | . . . 4 ⊢ (∃𝑧 ∈ 𝐴 𝑤 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
3 | rexcom4 2683 | . . . . 5 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
4 | rexcom4 2683 | . . . . . . 7 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦∃𝑧 ∈ 𝐴 (𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
5 | r19.42v 2565 | . . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐴 (𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) | |
6 | 5 | exbii 1569 | . . . . . . 7 ⊢ (∃𝑦∃𝑧 ∈ 𝐴 (𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) |
7 | 4, 6 | bitri 183 | . . . . . 6 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) |
8 | 7 | exbii 1569 | . . . . 5 ⊢ (∃𝑥∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) |
9 | 3, 8 | bitri 183 | . . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) |
10 | 2, 9 | bitri 183 | . . 3 ⊢ (∃𝑧 ∈ 𝐴 𝑤 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) |
11 | 10 | abbii 2233 | . 2 ⊢ {𝑤 ∣ ∃𝑧 ∈ 𝐴 𝑤 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)} |
12 | df-iun 3785 | . 2 ⊢ ∪ 𝑧 ∈ 𝐴 {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑧 ∈ 𝐴 𝑤 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}} | |
13 | df-opab 3960 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)} | |
14 | 11, 12, 13 | 3eqtr4i 2148 | 1 ⊢ ∪ 𝑧 ∈ 𝐴 {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1316 ∃wex 1453 ∈ wcel 1465 {cab 2103 ∃wrex 2394 〈cop 3500 ∪ ciun 3783 {copab 3958 |
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-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 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-opab 3960 |
This theorem is referenced by: (None) |
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