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Theorem dmopab3 4649
Description: The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
Assertion
Ref Expression
dmopab3 (∀𝑥𝐴𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem dmopab3
StepHypRef Expression
1 df-ral 2364 . 2 (∀𝑥𝐴𝑦𝜑 ↔ ∀𝑥(𝑥𝐴 → ∃𝑦𝜑))
2 pm4.71 381 . . 3 ((𝑥𝐴 → ∃𝑦𝜑) ↔ (𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦𝜑)))
32albii 1404 . 2 (∀𝑥(𝑥𝐴 → ∃𝑦𝜑) ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦𝜑)))
4 dmopab 4647 . . . . 5 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝜑)}
5 19.42v 1834 . . . . . 6 (∃𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦𝜑))
65abbii 2203 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)}
74, 6eqtri 2108 . . . 4 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)}
87eqeq1i 2095 . . 3 (dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴 ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} = 𝐴)
9 eqcom 2090 . . 3 (𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} = 𝐴)
10 abeq2 2196 . . 3 (𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦𝜑)))
118, 9, 103bitr2ri 207 . 2 (∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦𝜑)) ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴)
121, 3, 113bitri 204 1 (∀𝑥𝐴𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 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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