Theorem dmopab3 3318

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

Assertion

Ref	Expression
dmopab3	⊢ (∀x ∈ A ∃yφ ↔ dom {⟨x, y⟩∣(x ∈ A ⋀ φ)} = A)

Distinct variable group:   x,y,A

Proof of Theorem dmopab3

Step	Hyp	Ref	Expression
1		df-ral 1647	. 2 ⊢ (∀x ∈ A ∃yφ ↔ ∀x(x ∈ A → ∃yφ))
2		pm4.71 634	. . 3 ⊢ ((x ∈ A → ∃yφ) ↔ (x ∈ A ↔ (x ∈ A ⋀ ∃yφ)))
3	2	albii 998	. 2 ⊢ (∀x(x ∈ A → ∃yφ) ↔ ∀x(x ∈ A ↔ (x ∈ A ⋀ ∃yφ)))
4		dmopab 3316	. . . . 5 ⊢ dom {⟨x, y⟩∣(x ∈ A ⋀ φ)} = {x∣∃y(x ∈ A ⋀ φ)}
5		19.42v 1307	. . . . . 6 ⊢ (∃y(x ∈ A ⋀ φ) ↔ (x ∈ A ⋀ ∃yφ))
6	5	abbii 1573	. . . . 5 ⊢ {x∣∃y(x ∈ A ⋀ φ)} = {x∣(x ∈ A ⋀ ∃yφ)}
7	4, 6	eqtr 1493	. . . 4 ⊢ dom {⟨x, y⟩∣(x ∈ A ⋀ φ)} = {x∣(x ∈ A ⋀ ∃yφ)}
8	7	eqeq1i 1480	. . 3 ⊢ (dom {⟨x, y⟩∣(x ∈ A ⋀ φ)} = A ↔ {x∣(x ∈ A ⋀ ∃yφ)} = A)
9		eqcom 1475	. . 3 ⊢ (A = {x∣(x ∈ A ⋀ ∃yφ)} ↔ {x∣(x ∈ A ⋀ ∃yφ)} = A)
10		abeq2 1566	. . 3 ⊢ (A = {x∣(x ∈ A ⋀ ∃yφ)} ↔ ∀x(x ∈ A ↔ (x ∈ A ⋀ ∃yφ)))
11	8, 9, 10	3bitr2r 180	. 2 ⊢ (∀x(x ∈ A ↔ (x ∈ A ⋀ ∃yφ)) ↔ dom {⟨x, y⟩∣(x ∈ A ⋀ φ)} = A)
12	1, 3, 11	3bitr 177	1 ⊢ (∀x ∈ A ∃yφ ↔ dom {⟨x, y⟩∣(x ∈ A ⋀ φ)} = A)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 953   = wceq 955   ∈ wcel 957  ∃wex 979  {cab 1462  ∀wral 1643  {copab 2662  dom cdm 3166

This theorem is referenced by:  dmxp 3328  fnopabg 3611  fopab2 3818  dmrecpq 5057

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-opab 2663  df-dm 3184

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