Theorem relopabi 5721

Description: A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) Remove dependency on ax-sep 5218, ax-nul 5225, ax-pr 5347. (Revised by KP, 25-Oct-2021.)

Hypothesis

Ref	Expression
relopabi.1	⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Assertion

Ref	Expression
relopabi	⊢ Rel 𝐴

Proof of Theorem relopabi

Dummy variables 𝑧 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopabi.1	. . . . . . . 8 ⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		df-opab 5133	. . . . . . . 8 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
3	1, 2	eqtri 2766	. . . . . . 7 ⊢ 𝐴 = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4	3	abeq2i 2874	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5		simpl 482	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 = ⟨𝑥, 𝑦⟩)
6	5	2eximi 1839	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
7	4, 6	sylbi 216	. . . . 5 ⊢ (𝑧 ∈ 𝐴 → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
8		ax6evr 2019	. . . . . . . . . 10 ⊢ ∃𝑢 𝑦 = 𝑢
9		pm3.21 471	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → (𝑦 = 𝑢 → (𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧)))
10	9	eximdv 1921	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → (∃𝑢 𝑦 = 𝑢 → ∃𝑢(𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧)))
11	8, 10	mpi 20	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → ∃𝑢(𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧))
12		opeq2 4802	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑢 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩)
13		eqtr2 2762	. . . . . . . . . . . 12 ⊢ ((⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩ ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → ⟨𝑥, 𝑢⟩ = 𝑧)
14	13	eqcomd 2744	. . . . . . . . . . 11 ⊢ ((⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩ ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → 𝑧 = ⟨𝑥, 𝑢⟩)
15	12, 14	sylan 579	. . . . . . . . . 10 ⊢ ((𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → 𝑧 = ⟨𝑥, 𝑢⟩)
16	15	eximi 1838	. . . . . . . . 9 ⊢ (∃𝑢(𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
17	11, 16	syl 17	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
18	17	eqcoms 2746	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
19	18	2eximi 1839	. . . . . 6 ⊢ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑥∃𝑦∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
20		excomim 2165	. . . . . 6 ⊢ (∃𝑥∃𝑦∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑦∃𝑥∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
21	19, 20	syl 17	. . . . 5 ⊢ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑦∃𝑥∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
22		vex 3426	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
23		vex 3426	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
24	22, 23	pm3.2i 470	. . . . . . . . 9 ⊢ (𝑥 ∈ V ∧ 𝑢 ∈ V)
25	24	jctr 524	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑢⟩ → (𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
26	25	2eximi 1839	. . . . . . 7 ⊢ (∃𝑥∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑥∃𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
27		df-xp 5586	. . . . . . . . 9 ⊢ (V × V) = {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ V ∧ 𝑢 ∈ V)}
28		df-opab 5133	. . . . . . . . 9 ⊢ {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ V ∧ 𝑢 ∈ V)} = {𝑧 ∣ ∃𝑥∃𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V))}
29	27, 28	eqtri 2766	. . . . . . . 8 ⊢ (V × V) = {𝑧 ∣ ∃𝑥∃𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V))}
30	29	abeq2i 2874	. . . . . . 7 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑥∃𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
31	26, 30	sylibr 233	. . . . . 6 ⊢ (∃𝑥∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → 𝑧 ∈ (V × V))
32	31	eximi 1838	. . . . 5 ⊢ (∃𝑦∃𝑥∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑦 𝑧 ∈ (V × V))
33	7, 21, 32	3syl 18	. . . 4 ⊢ (𝑧 ∈ 𝐴 → ∃𝑦 𝑧 ∈ (V × V))
34		ax5e 1916	. . . 4 ⊢ (∃𝑦 𝑧 ∈ (V × V) → 𝑧 ∈ (V × V))
35	33, 34	syl 17	. . 3 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (V × V))
36	35	ssriv 3921	. 2 ⊢ 𝐴 ⊆ (V × V)
37		df-rel 5587	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
38	36, 37	mpbir 230	1 ⊢ Rel 𝐴

Syntax hints:   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715  Vcvv 3422   ⊆ wss 3883  ⟨cop 4564  {copab 5132   × cxp 5578  Rel wrel 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5133  df-xp 5586  df-rel 5587

This theorem is referenced by:  relopab  5723  erclwwlkrel  28282  erclwwlknrel  28331  relttrcl  33698