Theorem dmrab 31489

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⟩ → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
dmrab	⊢ dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ 𝐵 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

Step	Hyp	Ref	Expression
1		dmrab.1	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))
2	1	elrab 3648	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ 𝜓))
3		opelxp 5674	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
4	3	anbi1i 624	. . . . . . . 8 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ 𝜓) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓))
5		ancom 461	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
6	5	anbi1i 624	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓))
7	2, 4, 6	3bitri 296	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓))
8		anass 469	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)))
9		ancom 461	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝜓 ∧ 𝑥 ∈ 𝐴))
10	9	anbi2i 623	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝑦 ∈ 𝐵 ∧ (𝜓 ∧ 𝑥 ∈ 𝐴)))
11	7, 8, 10	3bitri 296	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (𝑦 ∈ 𝐵 ∧ (𝜓 ∧ 𝑥 ∈ 𝐴)))
12	11	exbii 1850	. . . . 5 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝜓 ∧ 𝑥 ∈ 𝐴)))
13		df-rex 3070	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝜓 ∧ 𝑥 ∈ 𝐴)))
14		r19.41v 3181	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑦 ∈ 𝐵 𝜓 ∧ 𝑥 ∈ 𝐴))
15	12, 13, 14	3bitr2i 298	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (∃𝑦 ∈ 𝐵 𝜓 ∧ 𝑥 ∈ 𝐴))
16	15	biancomi 463	. . 3 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜓))
17	16	abbii 2801	. 2 ⊢ {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑}} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜓)}
18		dfdm3 5848	. 2 ⊢ dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑}}
19		df-rab 3406	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ 𝐵 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜓)}
20	17, 18, 19	3eqtr4i 2769	1 ⊢ dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ 𝐵 𝜓}

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2708  ∃wrex 3069  {crab 3405  ⟨cop 4597   × cxp 5636  dom cdm 5638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-dm 5648

This theorem is referenced by:  fedgmullem2  32412