Theorem dmrab 33086

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 3645	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ 𝜓))
3		opelxp 5687	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
4	3	anbi1i 636	. . . . . . . 8 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ 𝜓) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓))
5		ancom 466	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
6	5	anbi1i 636	. . . . . . . 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 474	. . . . . . 7 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)))
9		ancom 466	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝜓 ∧ 𝑥 ∈ 𝐴))
10	9	anbi2i 635	. . . . . . 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 1881	. . . . 5 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝜓 ∧ 𝑥 ∈ 𝐴)))
13		df-rex 3088	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝜓 ∧ 𝑥 ∈ 𝐴)))
14		r19.41v 3193	. . . . 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 468	. . 3 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜓))
17	16	abbii 2828	. 2 ⊢ {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑}} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜓)}
18		dfdm3 5869	. 2 ⊢ dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑}}
19		df-rab 3414	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ 𝐵 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜓)}
20	17, 18, 19	3eqtr4i 2794	1 ⊢ dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ 𝐵 𝜓}

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739  ∃wrex 3087  {crab 3413  ⟨cop 4590   × cxp 5649  dom cdm 5651

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-dm 5661

This theorem is used by:  fedgmullem2  34255