Theorem dmopabss 5925

Description: Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem dmopabss

Step	Hyp	Ref	Expression
1		dmopab 5922	. 2 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)}
2		19.42v 1949	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑))
3	2	abbii 2798	. . 3 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)}
4		ssab2 4076	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)} ⊆ 𝐴
5	3, 4	eqsstri 4016	. 2 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
6	1, 5	eqsstri 4016	1 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴

Syntax hints:   ∧ wa 394  ∃wex 1773   ∈ wcel 2098  {cab 2705   ⊆ wss 3949  {copab 5214  dom cdm 5682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-dm 5692

This theorem is referenced by:  fvopab4ndm  7040  opabex  7238  pwfir  9209  perpln1  28542  dmadjss  31725  abrexdomjm  32331  abrexdom  37244