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Theorem dmrab 30410
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 3585 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ 𝜓))
3 opelxp 5555 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
43anbi1i 627 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ 𝜓) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝜓))
5 ancom 464 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
65anbi1i 627 . . . . . . . 8 (((𝑥𝐴𝑦𝐵) ∧ 𝜓) ↔ ((𝑦𝐵𝑥𝐴) ∧ 𝜓))
72, 4, 63bitri 300 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ ((𝑦𝐵𝑥𝐴) ∧ 𝜓))
8 anass 472 . . . . . . 7 (((𝑦𝐵𝑥𝐴) ∧ 𝜓) ↔ (𝑦𝐵 ∧ (𝑥𝐴𝜓)))
9 ancom 464 . . . . . . . 8 ((𝑥𝐴𝜓) ↔ (𝜓𝑥𝐴))
109anbi2i 626 . . . . . . 7 ((𝑦𝐵 ∧ (𝑥𝐴𝜓)) ↔ (𝑦𝐵 ∧ (𝜓𝑥𝐴)))
117, 8, 103bitri 300 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (𝑦𝐵 ∧ (𝜓𝑥𝐴)))
1211exbii 1854 . . . . 5 (∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ ∃𝑦(𝑦𝐵 ∧ (𝜓𝑥𝐴)))
13 df-rex 3059 . . . . 5 (∃𝑦𝐵 (𝜓𝑥𝐴) ↔ ∃𝑦(𝑦𝐵 ∧ (𝜓𝑥𝐴)))
14 r19.41v 3250 . . . . 5 (∃𝑦𝐵 (𝜓𝑥𝐴) ↔ (∃𝑦𝐵 𝜓𝑥𝐴))
1512, 13, 143bitr2i 302 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (∃𝑦𝐵 𝜓𝑥𝐴))
1615biancomi 466 . . 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 5724 . 2 dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑}}
19 df-rab 3062 . 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 209  wa 399   = wceq 1542  wex 1786  wcel 2113  {cab 2716  wrex 3054  {crab 3057  cop 4519   × cxp 5517  dom cdm 5519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-dif 3844  df-un 3846  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-br 5028  df-opab 5090  df-xp 5525  df-dm 5529
This theorem is referenced by:  fedgmullem2  31275
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