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Theorem dmrab 31737
Description: Domain of a restricted class abstraction over a cartesian product. (Contributed by Thierry Arnoux, 3-Jul-2023.)
Hypothesis
Ref Expression
dmrab.1 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
Assertion
Ref Expression
dmrab dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥𝐴 ∣ ∃𝑦𝐵 𝜓}
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝜑,𝑥,𝑦   𝜓,𝑧
Allowed substitution hints:   𝜑(𝑧)   𝜓(𝑥,𝑦)

Proof of Theorem dmrab
StepHypRef Expression
1 dmrab.1 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
21elrab 3684 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ 𝜓))
3 opelxp 5713 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
43anbi1i 625 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ 𝜓) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝜓))
5 ancom 462 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
65anbi1i 625 . . . . . . . 8 (((𝑥𝐴𝑦𝐵) ∧ 𝜓) ↔ ((𝑦𝐵𝑥𝐴) ∧ 𝜓))
72, 4, 63bitri 297 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ ((𝑦𝐵𝑥𝐴) ∧ 𝜓))
8 anass 470 . . . . . . 7 (((𝑦𝐵𝑥𝐴) ∧ 𝜓) ↔ (𝑦𝐵 ∧ (𝑥𝐴𝜓)))
9 ancom 462 . . . . . . . 8 ((𝑥𝐴𝜓) ↔ (𝜓𝑥𝐴))
109anbi2i 624 . . . . . . 7 ((𝑦𝐵 ∧ (𝑥𝐴𝜓)) ↔ (𝑦𝐵 ∧ (𝜓𝑥𝐴)))
117, 8, 103bitri 297 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (𝑦𝐵 ∧ (𝜓𝑥𝐴)))
1211exbii 1851 . . . . 5 (∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ ∃𝑦(𝑦𝐵 ∧ (𝜓𝑥𝐴)))
13 df-rex 3072 . . . . 5 (∃𝑦𝐵 (𝜓𝑥𝐴) ↔ ∃𝑦(𝑦𝐵 ∧ (𝜓𝑥𝐴)))
14 r19.41v 3189 . . . . 5 (∃𝑦𝐵 (𝜓𝑥𝐴) ↔ (∃𝑦𝐵 𝜓𝑥𝐴))
1512, 13, 143bitr2i 299 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (∃𝑦𝐵 𝜓𝑥𝐴))
1615biancomi 464 . . 3 (∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜓))
1716abbii 2803 . 2 {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑}} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜓)}
18 dfdm3 5888 . 2 dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑}}
19 df-rab 3434 . 2 {𝑥𝐴 ∣ ∃𝑦𝐵 𝜓} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜓)}
2017, 18, 193eqtr4i 2771 1 dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥𝐴 ∣ ∃𝑦𝐵 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wrex 3071  {crab 3433  cop 4635   × cxp 5675  dom cdm 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-dm 5687
This theorem is referenced by:  fedgmullem2  32715
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