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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 4013 | . . . . 5 ⊢ (𝑤 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
2 | 1 | rexbii 2373 | . . . 4 ⊢ (∃𝑧 ∈ 𝐴 𝑤 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
3 | rexcom4 2622 | . . . . 5 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
4 | rexcom4 2622 | . . . . . . 7 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦∃𝑧 ∈ 𝐴 (𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
5 | r19.42v 2511 | . . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐴 (𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) | |
6 | 5 | exbii 1536 | . . . . . . 7 ⊢ (∃𝑦∃𝑧 ∈ 𝐴 (𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) |
7 | 4, 6 | bitri 182 | . . . . . 6 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) |
8 | 7 | exbii 1536 | . . . . 5 ⊢ (∃𝑥∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) |
9 | 3, 8 | bitri 182 | . . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) |
10 | 2, 9 | bitri 182 | . . 3 ⊢ (∃𝑧 ∈ 𝐴 𝑤 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) |
11 | 10 | abbii 2194 | . 2 ⊢ {𝑤 ∣ ∃𝑧 ∈ 𝐴 𝑤 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)} |
12 | df-iun 3680 | . 2 ⊢ ∪ 𝑧 ∈ 𝐴 {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑧 ∈ 𝐴 𝑤 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}} | |
13 | df-opab 3840 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)} | |
14 | 11, 12, 13 | 3eqtr4i 2111 | 1 ⊢ ∪ 𝑧 ∈ 𝐴 {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 = wceq 1284 ∃wex 1421 ∈ wcel 1433 {cab 2067 ∃wrex 2349 〈cop 3401 ∪ ciun 3678 {copab 3838 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-iun 3680 df-opab 3840 |
This theorem is referenced by: (None) |
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