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Theorem dmrab 30844
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 3624 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ 𝜓))
3 opelxp 5625 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
43anbi1i 624 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ 𝜓) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝜓))
5 ancom 461 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
65anbi1i 624 . . . . . . . 8 (((𝑥𝐴𝑦𝐵) ∧ 𝜓) ↔ ((𝑦𝐵𝑥𝐴) ∧ 𝜓))
72, 4, 63bitri 297 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ ((𝑦𝐵𝑥𝐴) ∧ 𝜓))
8 anass 469 . . . . . . 7 (((𝑦𝐵𝑥𝐴) ∧ 𝜓) ↔ (𝑦𝐵 ∧ (𝑥𝐴𝜓)))
9 ancom 461 . . . . . . . 8 ((𝑥𝐴𝜓) ↔ (𝜓𝑥𝐴))
109anbi2i 623 . . . . . . 7 ((𝑦𝐵 ∧ (𝑥𝐴𝜓)) ↔ (𝑦𝐵 ∧ (𝜓𝑥𝐴)))
117, 8, 103bitri 297 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (𝑦𝐵 ∧ (𝜓𝑥𝐴)))
1211exbii 1850 . . . . 5 (∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ ∃𝑦(𝑦𝐵 ∧ (𝜓𝑥𝐴)))
13 df-rex 3070 . . . . 5 (∃𝑦𝐵 (𝜓𝑥𝐴) ↔ ∃𝑦(𝑦𝐵 ∧ (𝜓𝑥𝐴)))
14 r19.41v 3276 . . . . 5 (∃𝑦𝐵 (𝜓𝑥𝐴) ↔ (∃𝑦𝐵 𝜓𝑥𝐴))
1512, 13, 143bitr2i 299 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (∃𝑦𝐵 𝜓𝑥𝐴))
1615biancomi 463 . . 3 (∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜓))
1716abbii 2808 . 2 {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑}} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜓)}
18 dfdm3 5796 . 2 dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑}}
19 df-rab 3073 . 2 {𝑥𝐴 ∣ ∃𝑦𝐵 𝜓} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜓)}
2017, 18, 193eqtr4i 2776 1 dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥𝐴 ∣ ∃𝑦𝐵 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wex 1782  wcel 2106  {cab 2715  wrex 3065  {crab 3068  cop 4567   × cxp 5587  dom cdm 5589
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-dm 5599
This theorem is referenced by:  fedgmullem2  31711
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