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Theorem dmopab3 5912
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 3056 . 2 (∀𝑥𝐴𝑦𝜑 ↔ ∀𝑥(𝑥𝐴 → ∃𝑦𝜑))
2 pm4.71 557 . . 3 ((𝑥𝐴 → ∃𝑦𝜑) ↔ (𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦𝜑)))
32albii 1813 . 2 (∀𝑥(𝑥𝐴 → ∃𝑦𝜑) ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦𝜑)))
4 dmopab 5908 . . . . 5 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝜑)}
5 19.42v 1949 . . . . . 6 (∃𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦𝜑))
65abbii 2796 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)}
74, 6eqtri 2754 . . . 4 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)}
87eqeq1i 2731 . . 3 (dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴 ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} = 𝐴)
9 eqcom 2733 . . 3 (𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} = 𝐴)
10 eqabb 2867 . . 3 (𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦𝜑)))
118, 9, 103bitr2ri 300 . 2 (∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦𝜑)) ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴)
121, 3, 113bitri 297 1 (∀𝑥𝐴𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1531   = wceq 1533  wex 1773  wcel 2098  {cab 2703  wral 3055  {copab 5203  dom cdm 5669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-dm 5679
This theorem is referenced by:  dmxp  5921  fnopabg  6680  opabn1stprc  8040  n0el2  37714
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