Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3995 | . . . . 5 ⊢ (𝑤 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
2 | 1 | rexbii 2331 | . . . 4 ⊢ (∃𝑧 ∈ 𝐴 𝑤 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
3 | rexcom4 2577 | . . . . 5 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
4 | rexcom4 2577 | . . . . . . 7 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦∃𝑧 ∈ 𝐴 (𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
5 | r19.42v 2467 | . . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐴 (𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) | |
6 | 5 | exbii 1496 | . . . . . . 7 ⊢ (∃𝑦∃𝑧 ∈ 𝐴 (𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) |
7 | 4, 6 | bitri 173 | . . . . . 6 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) |
8 | 7 | exbii 1496 | . . . . 5 ⊢ (∃𝑥∃𝑧 ∈ 𝐴 ∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) |
9 | 3, 8 | bitri 173 | . . . 4 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) |
10 | 2, 9 | bitri 173 | . . 3 ⊢ (∃𝑧 ∈ 𝐴 𝑤 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)) |
11 | 10 | abbii 2153 | . 2 ⊢ {𝑤 ∣ ∃𝑧 ∈ 𝐴 𝑤 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)} |
12 | df-iun 3659 | . 2 ⊢ ∪ 𝑧 ∈ 𝐴 {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑧 ∈ 𝐴 𝑤 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}} | |
13 | df-opab 3819 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ ∃𝑧 ∈ 𝐴 𝜑)} | |
14 | 11, 12, 13 | 3eqtr4i 2070 | 1 ⊢ ∪ 𝑧 ∈ 𝐴 {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 ∃wex 1381 ∈ wcel 1393 {cab 2026 ∃wrex 2307 〈cop 3378 ∪ ciun 3657 {copab 3817 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-iun 3659 df-opab 3819 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |