Theorem dmopab 5889

Description: The domain of a class of ordered pairs. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)

Assertion

Ref	Expression
dmopab	⊢ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem dmopab

Step	Hyp	Ref	Expression
1		nfopab1 5169	. . 3 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		nfopab2 5170	. . 3 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
3	1, 2	dfdmf 5870	. 2 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦}
4		df-br 5100	. . . . 5 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5		opabidw 5493	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
6	4, 5	bitri 277	. . . 4 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ 𝜑)
7	6	exbii 1867	. . 3 ⊢ (∃𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ∃𝑦𝜑)
8	7	abbii 2828	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦} = {𝑥 ∣ ∃𝑦𝜑}
9	3, 8	eqtri 2784	1 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑}

Syntax hints:   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739  ⟨cop 4587   class class class wbr 5099  {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:  dmopabelb  5890  dmopabss  5892  dmopab3  5893  mptfnf  6652  opabiotadm  6944  fndmin  7022  dmoprab  7495  zfrep6OLD  7932  hartogslem1  9487  dmttrcl  9673  rankf  9749  dfac3  10074  axdc2lem  10402  shftdm  15081  dfiso2  17788  adjeu  32038  satfdm  35683  fmla0  35696  fmlasuc0  35698