Theorem relopabi 5796

Description: A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) Remove dependency on ax-sep 5247, ax-nul 5257, ax-pr 5391. (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 5164	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
3	1, 2	eqtri 2786	. . . . . 6 ⊢ 𝐴 = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4	3	eqabri 2905	. . . . 5 ⊢ (𝑧 ∈ 𝐴 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5		simpl 486	. . . . . 6 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 = ⟨𝑥, 𝑦⟩)
6	5	2eximi 1857	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
7	4, 6	sylbi 219	. . . 4 ⊢ (𝑧 ∈ 𝐴 → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
8		ax6evr 2036	. . . . . . . . 9 ⊢ ∃𝑢 𝑦 = 𝑢
9		pm3.21 475	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → (𝑦 = 𝑢 → (𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧)))
10	9	eximdv 1938	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → (∃𝑢 𝑦 = 𝑢 → ∃𝑢(𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧)))
11	8, 10	mpi 20	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → ∃𝑢(𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧))
12		opeq2 4833	. . . . . . . . . 10 ⊢ (𝑦 = 𝑢 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩)
13		eqtr2 2784	. . . . . . . . . . 11 ⊢ ((⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩ ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → ⟨𝑥, 𝑢⟩ = 𝑧)
14	13	eqcomd 2769	. . . . . . . . . 10 ⊢ ((⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩ ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → 𝑧 = ⟨𝑥, 𝑢⟩)
15	12, 14	sylan 589	. . . . . . . . 9 ⊢ ((𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → 𝑧 = ⟨𝑥, 𝑢⟩)
16	15	eximi 1856	. . . . . . . 8 ⊢ (∃𝑢(𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
17	11, 16	syl 17	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
18	17	eqcoms 2771	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
19	18	2eximi 1857	. . . . 5 ⊢ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑥∃𝑦∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
20		excomim 2198	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑦∃𝑥∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
21	19, 20	syl 17	. . . 4 ⊢ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑦∃𝑥∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
22		vex 3459	. . . . . . . . 9 ⊢ 𝑥 ∈ V
23		vex 3459	. . . . . . . . 9 ⊢ 𝑢 ∈ V
24	22, 23	pm3.2i 474	. . . . . . . 8 ⊢ (𝑥 ∈ V ∧ 𝑢 ∈ V)
25	24	jctr 532	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑢⟩ → (𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
26	25	2eximi 1857	. . . . . 6 ⊢ (∃𝑥∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑥∃𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
27		df-xp 5654	. . . . . . . 8 ⊢ (V × V) = {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ V ∧ 𝑢 ∈ V)}
28		df-opab 5164	. . . . . . . 8 ⊢ {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ V ∧ 𝑢 ∈ V)} = {𝑧 ∣ ∃𝑥∃𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V))}
29	27, 28	eqtri 2786	. . . . . . 7 ⊢ (V × V) = {𝑧 ∣ ∃𝑥∃𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V))}
30	29	eqabri 2905	. . . . . 6 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑥∃𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
31	26, 30	sylibr 236	. . . . 5 ⊢ (∃𝑥∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → 𝑧 ∈ (V × V))
32	31	eximi 1856	. . . 4 ⊢ (∃𝑦∃𝑥∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑦 𝑧 ∈ (V × V))
33		ax5e 1933	. . . 4 ⊢ (∃𝑦 𝑧 ∈ (V × V) → 𝑧 ∈ (V × V))
34	7, 21, 32, 33	4syl 19	. . 3 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (V × V))
35	34	ssriv 3941	. 2 ⊢ 𝐴 ⊆ (V × V)
36		df-rel 5655	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
37	35, 36	mpbir 233	1 ⊢ Rel 𝐴

Syntax hints:   ∧ wa 399   = wceq 1561  ∃wex 1800   ∈ wcel 2143  {cab 2741  Vcvv 3455   ⊆ wss 3905  ⟨cop 4589  {copab 5163   × cxp 5646  Rel wrel 5653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-opab 5164  df-xp 5654  df-rel 5655

This theorem is referenced by:  relopab  5798  relttrcl  9668  erclwwlkrel  30220  erclwwlknrel  30269  relsucmap  38967