Theorem dmopabss 5892

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 5889	. 2 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)}
2		19.42v 1972	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑))
3	2	abbii 2828	. . 3 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)}
4		ssab2 4032	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑)} ⊆ 𝐴
5	3, 4	eqsstri 3982	. 2 ⊢ {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
6	1, 5	eqsstri 3982	1 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴

Syntax hints:   ∧ wa 399  ∃wex 1798   ∈ wcel 2141  {cab 2739   ⊆ wss 3904  {copab 5161  dom cdm 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-dm 5655

This theorem is referenced by:  fvopab4ndm  7002  opabex  7200  pwfir  9257  perpln1  28856  dmadjss  32036  abrexdomjm  32655  fineqvnttrclse  35384  abrexdom  38193  modelaxreplem2  45519