Theorem relopabi 5810

Description: A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) Remove dependency on ax-sep 5261, ax-nul 5271, ax-pr 5405. (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	. . . . . . 7 ⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		df-opab 5178	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
3	1, 2	eqtri 2792	. . . . . 6 ⊢ 𝐴 = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4	3	eqabri 2911	. . . . 5 ⊢ (𝑧 ∈ 𝐴 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5		simpl 487	. . . . . 6 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 = ⟨𝑥, 𝑦⟩)
6	5	2eximi 1863	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
7	4, 6	sylbi 220	. . . 4 ⊢ (𝑧 ∈ 𝐴 → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
8		ax6evr 2042	. . . . . . . . 9 ⊢ ∃𝑢 𝑦 = 𝑢
9		pm3.21 476	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → (𝑦 = 𝑢 → (𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧)))
10	9	eximdv 1944	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → (∃𝑢 𝑦 = 𝑢 → ∃𝑢(𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧)))
11	8, 10	mpi 21	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → ∃𝑢(𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧))
12		opeq2 4843	. . . . . . . . . 10 ⊢ (𝑦 = 𝑢 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩)
13		eqtr2 2790	. . . . . . . . . . 11 ⊢ ((⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩ ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → ⟨𝑥, 𝑢⟩ = 𝑧)
14	13	eqcomd 2775	. . . . . . . . . 10 ⊢ ((⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩ ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → 𝑧 = ⟨𝑥, 𝑢⟩)
15	12, 14	sylan 591	. . . . . . . . 9 ⊢ ((𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → 𝑧 = ⟨𝑥, 𝑢⟩)
16	15	eximi 1862	. . . . . . . 8 ⊢ (∃𝑢(𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
17	11, 16	syl 18	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
18	17	eqcoms 2777	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
19	18	2eximi 1863	. . . . 5 ⊢ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑥∃𝑦∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
20		excomim 2204	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑦∃𝑥∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
21	19, 20	syl 18	. . . 4 ⊢ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑦∃𝑥∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
22		vex 3467	. . . . . . . . 9 ⊢ 𝑥 ∈ V
23		vex 3467	. . . . . . . . 9 ⊢ 𝑢 ∈ V
24	22, 23	pm3.2i 475	. . . . . . . 8 ⊢ (𝑥 ∈ V ∧ 𝑢 ∈ V)
25	24	jctr 533	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑢⟩ → (𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
26	25	2eximi 1863	. . . . . 6 ⊢ (∃𝑥∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑥∃𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
27		df-xp 5668	. . . . . . . 8 ⊢ (V × V) = {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ V ∧ 𝑢 ∈ V)}
28		df-opab 5178	. . . . . . . 8 ⊢ {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ V ∧ 𝑢 ∈ V)} = {𝑧 ∣ ∃𝑥∃𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V))}
29	27, 28	eqtri 2792	. . . . . . 7 ⊢ (V × V) = {𝑧 ∣ ∃𝑥∃𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V))}
30	29	eqabri 2911	. . . . . 6 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑥∃𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
31	26, 30	sylibr 237	. . . . 5 ⊢ (∃𝑥∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → 𝑧 ∈ (V × V))
32	31	eximi 1862	. . . 4 ⊢ (∃𝑦∃𝑥∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑦 𝑧 ∈ (V × V))
33		ax5e 1939	. . . 4 ⊢ (∃𝑦 𝑧 ∈ (V × V) → 𝑧 ∈ (V × V))
34	7, 21, 32, 33	4syl 20	. . 3 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (V × V))
35	34	ssriv 3949	. 2 ⊢ 𝐴 ⊆ (V × V)
36		df-rel 5669	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
37	35, 36	mpbir 234	1 ⊢ Rel 𝐴

Syntax hints:   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  Vcvv 3463   ⊆ wss 3913  ⟨cop 4600  {copab 5177   × cxp 5660  Rel wrel 5667

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-11 2198  ax-12 2219  ax-ext 2741

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-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-opab 5178  df-xp 5668  df-rel 5669

This theorem is referenced by:  relopab  5812  relttrcl  9681  erclwwlkrel  30309  erclwwlknrel  30358  relsucmap  39040