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Mirrors > Home > ILE Home > Th. List > dmopab3 | GIF version |
Description: The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.) |
Ref | Expression |
---|---|
dmopab3 | ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2364 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦𝜑)) | |
2 | pm4.71 381 | . . 3 ⊢ ((𝑥 ∈ 𝐴 → ∃𝑦𝜑) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑))) | |
3 | 2 | albii 1404 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∃𝑦𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑))) |
4 | dmopab 4647 | . . . . 5 ⊢ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)} | |
5 | 19.42v 1834 | . . . . . 6 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)) | |
6 | 5 | abbii 2203 | . . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)} |
7 | 4, 6 | eqtri 2108 | . . . 4 ⊢ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)} |
8 | 7 | eqeq1i 2095 | . . 3 ⊢ (dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴 ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)} = 𝐴) |
9 | eqcom 2090 | . . 3 ⊢ (𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)} ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)} = 𝐴) | |
10 | abeq2 2196 | . . 3 ⊢ (𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑))) | |
11 | 8, 9, 10 | 3bitr2ri 207 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)) ↔ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴) |
12 | 1, 3, 11 | 3bitri 204 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1287 = wceq 1289 ∃wex 1426 ∈ wcel 1438 {cab 2074 ∀wral 2359 {copab 3898 dom cdm 4438 |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-br 3846 df-opab 3900 df-dm 4448 |
This theorem is referenced by: dmxpm 4656 fnopabg 5137 |
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