Theorem dmopabss 2540

Description: Upper bound for the domain of a restricted class of ordered pairs.

Assertion

Ref	Expression
dmopabss	⊢ dom {⟨x, y⟩∣(x ∈ A ∧ φ)} ⊆ A

Distinct variable group(s):   x,y,A

Proof of Theorem dmopabss

Step	Hyp	Ref	Expression
1		dmopab 2539	. 2 ⊢ dom {⟨x, y⟩∣(x ∈ A ∧ φ)} = {x∣∃y(x ∈ A ∧ φ)}
2		19.42v 966	. . . 4 ⊢ (∃y(x ∈ A ∧ φ) ↔ (x ∈ A ∧ ∃yφ))
3	2	biabi 1181	. . 3 ⊢ {x∣∃y(x ∈ A ∧ φ)} = {x∣(x ∈ A ∧ ∃yφ)}
4		ssab 1555	. . 3 ⊢ {x∣(x ∈ A ∧ ∃yφ)} ⊆ A
5	3, 4	eqsstr 1530	. 2 ⊢ {x∣∃y(x ∈ A ∧ φ)} ⊆ A
6	1, 5	eqsstr 1530	1 ⊢ dom {⟨x, y⟩∣(x ∈ A ∧ φ)} ⊆ A

Syntax hints:   ∧ wa 196  ∃wex 678  {cab 1090   ∈ wcel 1092   ⊆ wss 1487  {copab 2055  dom cdm 2410

This theorem is referenced by:  funopabex 2742  qsex 3231

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-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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