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Theorem dmrab 29948
 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 3621 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ 𝜓))
3 opelxp 5486 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
43anbi1i 623 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ 𝜓) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝜓))
5 ancom 461 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) ↔ (𝑦𝐵𝑥𝐴))
65anbi1i 623 . . . . . . . 8 (((𝑥𝐴𝑦𝐵) ∧ 𝜓) ↔ ((𝑦𝐵𝑥𝐴) ∧ 𝜓))
72, 4, 63bitri 298 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ ((𝑦𝐵𝑥𝐴) ∧ 𝜓))
8 anass 469 . . . . . . 7 (((𝑦𝐵𝑥𝐴) ∧ 𝜓) ↔ (𝑦𝐵 ∧ (𝑥𝐴𝜓)))
9 ancom 461 . . . . . . . 8 ((𝑥𝐴𝜓) ↔ (𝜓𝑥𝐴))
109anbi2i 622 . . . . . . 7 ((𝑦𝐵 ∧ (𝑥𝐴𝜓)) ↔ (𝑦𝐵 ∧ (𝜓𝑥𝐴)))
117, 8, 103bitri 298 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (𝑦𝐵 ∧ (𝜓𝑥𝐴)))
1211exbii 1833 . . . . 5 (∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ ∃𝑦(𝑦𝐵 ∧ (𝜓𝑥𝐴)))
13 df-rex 3113 . . . . 5 (∃𝑦𝐵 (𝜓𝑥𝐴) ↔ ∃𝑦(𝑦𝐵 ∧ (𝜓𝑥𝐴)))
14 r19.41v 3310 . . . . 5 (∃𝑦𝐵 (𝜓𝑥𝐴) ↔ (∃𝑦𝐵 𝜓𝑥𝐴))
1512, 13, 143bitr2i 300 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (∃𝑦𝐵 𝜓𝑥𝐴))
1615biancomi 463 . . 3 (∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜓))
1716abbii 2863 . 2 {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑}} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜓)}
18 dfdm3 5651 . 2 dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑}}
19 df-rab 3116 . 2 {𝑥𝐴 ∣ ∃𝑦𝐵 𝜓} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜓)}
2017, 18, 193eqtr4i 2831 1 dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥𝐴 ∣ ∃𝑦𝐵 𝜓}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1525  ∃wex 1765   ∈ wcel 2083  {cab 2777  ∃wrex 3108  {crab 3111  ⟨cop 4484   × cxp 5448  dom cdm 5450 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-xp 5456  df-dm 5460 This theorem is referenced by:  fedgmullem2  30626
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