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Theorem dmopab3 5756
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 3114 . 2 (∀𝑥𝐴𝑦𝜑 ↔ ∀𝑥(𝑥𝐴 → ∃𝑦𝜑))
2 pm4.71 561 . . 3 ((𝑥𝐴 → ∃𝑦𝜑) ↔ (𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦𝜑)))
32albii 1821 . 2 (∀𝑥(𝑥𝐴 → ∃𝑦𝜑) ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦𝜑)))
4 dmopab 5752 . . . . 5 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝜑)}
5 19.42v 1954 . . . . . 6 (∃𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦𝜑))
65abbii 2866 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)}
74, 6eqtri 2824 . . . 4 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)}
87eqeq1i 2806 . . 3 (dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴 ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} = 𝐴)
9 eqcom 2808 . . 3 (𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} = 𝐴)
10 abeq2 2925 . . 3 (𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦𝜑)))
118, 9, 103bitr2ri 303 . 2 (∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦𝜑)) ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴)
121, 3, 113bitri 300 1 (∀𝑥𝐴𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  wcel 2112  {cab 2779  wral 3109  {copab 5095  dom cdm 5523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-dm 5533
This theorem is referenced by:  dmxp  5767  fnopabg  6461  opabn1stprc  7742  n0el2  35749
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