Theorem dmrab 32780

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 3659	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ 𝜓))
3		opelxp 5695	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
4	3	anbi1i 635	. . . . . . . 8 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ 𝜓) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓))
5		ancom 465	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
6	5	anbi1i 635	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓))
7	2, 4, 6	3bitri 300	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓))
8		anass 473	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)))
9		ancom 465	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝜓 ∧ 𝑥 ∈ 𝐴))
10	9	anbi2i 634	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝑦 ∈ 𝐵 ∧ (𝜓 ∧ 𝑥 ∈ 𝐴)))
11	7, 8, 10	3bitri 300	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (𝑦 ∈ 𝐵 ∧ (𝜓 ∧ 𝑥 ∈ 𝐴)))
12	11	exbii 1875	. . . . 5 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝜓 ∧ 𝑥 ∈ 𝐴)))
13		df-rex 3096	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝜓 ∧ 𝑥 ∈ 𝐴)))
14		r19.41v 3201	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑦 ∈ 𝐵 𝜓 ∧ 𝑥 ∈ 𝐴))
15	12, 13, 14	3bitr2i 302	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (∃𝑦 ∈ 𝐵 𝜓 ∧ 𝑥 ∈ 𝐴))
16	15	biancomi 467	. . 3 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜓))
17	16	abbii 2836	. 2 ⊢ {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑}} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜓)}
18		dfdm3 5875	. 2 ⊢ dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑}}
19		df-rab 3424	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ 𝐵 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜓)}
20	17, 18, 19	3eqtr4i 2802	1 ⊢ dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ 𝐵 𝜓}

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  ∃wrex 3095  {crab 3423  ⟨cop 4597   × cxp 5657  dom cdm 5659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-xp 5665  df-dm 5669

This theorem is referenced by:  fedgmullem2  33961