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Theorem dmopab3 4722
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 2398 . 2 (∀𝑥𝐴𝑦𝜑 ↔ ∀𝑥(𝑥𝐴 → ∃𝑦𝜑))
2 pm4.71 386 . . 3 ((𝑥𝐴 → ∃𝑦𝜑) ↔ (𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦𝜑)))
32albii 1431 . 2 (∀𝑥(𝑥𝐴 → ∃𝑦𝜑) ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦𝜑)))
4 dmopab 4720 . . . . 5 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝜑)}
5 19.42v 1862 . . . . . 6 (∃𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦𝜑))
65abbii 2233 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)}
74, 6eqtri 2138 . . . 4 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)}
87eqeq1i 2125 . . 3 (dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴 ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} = 𝐴)
9 eqcom 2119 . . 3 (𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} ↔ {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} = 𝐴)
10 abeq2 2226 . . 3 (𝐴 = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝜑)} ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦𝜑)))
118, 9, 103bitr2ri 208 . 2 (∀𝑥(𝑥𝐴 ↔ (𝑥𝐴 ∧ ∃𝑦𝜑)) ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴)
121, 3, 113bitri 205 1 (∀𝑥𝐴𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1314   = wceq 1316  wex 1453  wcel 1465  {cab 2103  wral 2393  {copab 3958  dom cdm 4509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-dm 4519
This theorem is referenced by:  dmxpm  4729  dmxpid  4730  fnopabg  5216  acfun  7031  ccfunen  7047
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