Theorem dmopab2 2541

Description: The domain of a restricted class of ordered pairs.

Assertion

Ref	Expression
dmopab2	⊢ (∀x ∈ A ∃yφ ↔ dom {⟨x, y⟩∣(x ∈ A ∧ φ)} = A)

Distinct variable group(s):   x,y,A

Proof of Theorem dmopab2

Step	Hyp	Ref	Expression
1		df-ral 1205	. 2 ⊢ (∀x ∈ A ∃yφ ↔ ∀x(x ∈ A → ∃yφ))
2		pm4.71 481	. . 3 ⊢ ((x ∈ A → ∃yφ) ↔ (x ∈ A ↔ (x ∈ A ∧ ∃yφ)))
3	2	bial 695	. 2 ⊢ (∀x(x ∈ A → ∃yφ) ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ ∃yφ)))
4		cleqab 1174	. . 3 ⊢ (A = {x∣(x ∈ A ∧ ∃yφ)} ↔ ∀x(x ∈ A ↔ (x ∈ A ∧ ∃yφ)))
5		cleqcom 1103	. . . 4 ⊢ (A = {x∣(x ∈ A ∧ ∃yφ)} ↔ {x∣(x ∈ A ∧ ∃yφ)} = A)
6		dmopab 2539	. . . . . 6 ⊢ dom {⟨x, y⟩∣(x ∈ A ∧ φ)} = {x∣∃y(x ∈ A ∧ φ)}
7		19.42v 966	. . . . . . 7 ⊢ (∃y(x ∈ A ∧ φ) ↔ (x ∈ A ∧ ∃yφ))
8	7	biabi 1181	. . . . . 6 ⊢ {x∣∃y(x ∈ A ∧ φ)} = {x∣(x ∈ A ∧ ∃yφ)}
9	6, 8	eqtr 1119	. . . . 5 ⊢ dom {⟨x, y⟩∣(x ∈ A ∧ φ)} = {x∣(x ∈ A ∧ ∃yφ)}
10	9	cleq1i 1108	. . . 4 ⊢ (dom {⟨x, y⟩∣(x ∈ A ∧ φ)} = A ↔ {x∣(x ∈ A ∧ ∃yφ)} = A)
11	5, 10	bitr4 154	. . 3 ⊢ (A = {x∣(x ∈ A ∧ ∃yφ)} ↔ dom {⟨x, y⟩∣(x ∈ A ∧ φ)} = A)
12	4, 11	bitr3 153	. 2 ⊢ (∀x(x ∈ A ↔ (x ∈ A ∧ ∃yφ)) ↔ dom {⟨x, y⟩∣(x ∈ A ∧ φ)} = A)
13	1, 3, 12	3bitr 155	1 ⊢ (∀x ∈ A ∃yφ ↔ dom {⟨x, y⟩∣(x ∈ A ∧ φ)} = A)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  {copab 2055  dom cdm 2410

This theorem is referenced by:  dmxp 2552  fnopabg 2745  fopab2 2891  dmrecpq 3868

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-dm 2428

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