Theorem relopabi 5694

Description: A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) Remove dependency on ax-sep 5203, ax-nul 5210, ax-pr 5330. (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 5129	. . . . . . . 8 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
3	1, 2	eqtri 2844	. . . . . . 7 ⊢ 𝐴 = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4	3	abeq2i 2948	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5		simpl 485	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 = ⟨𝑥, 𝑦⟩)
6	5	2eximi 1836	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
7	4, 6	sylbi 219	. . . . 5 ⊢ (𝑧 ∈ 𝐴 → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
8		ax6evr 2022	. . . . . . . . . 10 ⊢ ∃𝑢 𝑦 = 𝑢
9		pm3.21 474	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ = 𝑧 → (𝑦 = 𝑢 → (𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧)))
10	9	eximdv 1918	. . . . . . . . . 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 4804	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑢 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩)
13		eqtr2 2842	. . . . . . . . . . . 12 ⊢ ((⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩ ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → ⟨𝑥, 𝑢⟩ = 𝑧)
14	13	eqcomd 2827	. . . . . . . . . . 11 ⊢ ((⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩ ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → 𝑧 = ⟨𝑥, 𝑢⟩)
15	12, 14	sylan 582	. . . . . . . . . 10 ⊢ ((𝑦 = 𝑢 ∧ ⟨𝑥, 𝑦⟩ = 𝑧) → 𝑧 = ⟨𝑥, 𝑢⟩)
16	15	eximi 1835	. . . . . . . . 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 2829	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
19	18	2eximi 1836	. . . . . 6 ⊢ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑥∃𝑦∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩)
20		excomim 2170	. . . . . 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 3497	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
23		vex 3497	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
24	22, 23	pm3.2i 473	. . . . . . . . 9 ⊢ (𝑥 ∈ V ∧ 𝑢 ∈ V)
25	24	jctr 527	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑢⟩ → (𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
26	25	2eximi 1836	. . . . . . 7 ⊢ (∃𝑥∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → ∃𝑥∃𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
27		df-xp 5561	. . . . . . . . 9 ⊢ (V × V) = {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ V ∧ 𝑢 ∈ V)}
28		df-opab 5129	. . . . . . . . 9 ⊢ {⟨𝑥, 𝑢⟩ ∣ (𝑥 ∈ V ∧ 𝑢 ∈ V)} = {𝑧 ∣ ∃𝑥∃𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V))}
29	27, 28	eqtri 2844	. . . . . . . 8 ⊢ (V × V) = {𝑧 ∣ ∃𝑥∃𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V))}
30	29	abeq2i 2948	. . . . . . 7 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑥∃𝑢(𝑧 = ⟨𝑥, 𝑢⟩ ∧ (𝑥 ∈ V ∧ 𝑢 ∈ V)))
31	26, 30	sylibr 236	. . . . . 6 ⊢ (∃𝑥∃𝑢 𝑧 = ⟨𝑥, 𝑢⟩ → 𝑧 ∈ (V × V))
32	31	eximi 1835	. . . . 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 1913	. . . 4 ⊢ (∃𝑦 𝑧 ∈ (V × V) → 𝑧 ∈ (V × V))
35	33, 34	syl 17	. . 3 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (V × V))
36	35	ssriv 3971	. 2 ⊢ 𝐴 ⊆ (V × V)
37		df-rel 5562	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
38	36, 37	mpbir 233	1 ⊢ Rel 𝐴

Syntax hints:   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2799  Vcvv 3494   ⊆ wss 3936  ⟨cop 4573  {copab 5128   × cxp 5553  Rel wrel 5560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-opab 5129  df-xp 5561  df-rel 5562

This theorem is referenced by:  relopab  5696  mptrel  5697  reli  5698  rele  5699  relcnv  5967  cotrg  5971  relco  6097  brfvopabrbr  6765  reloprab  7213  reldmoprab  7259  relrpss  7450  eqer  8324  ecopover  8401  relen  8514  reldom  8515  relfsupp  8835  relwdom  9030  fpwwe2lem2  10054  fpwwe2lem3  10055  fpwwe2lem6  10057  fpwwe2lem7  10058  fpwwe2lem9  10060  fpwwe2lem11  10062  fpwwe2lem12  10063  fpwwe2lem13  10064  fpwwelem  10067  climrel  14849  rlimrel  14850  brstruct  16492  sscrel  17083  gaorber  18438  sylow2a  18744  efgrelexlemb  18876  efgcpbllemb  18881  rellindf  20952  2ndcctbss  22063  refrel  22116  vitalilem1  24209  lgsquadlem1  25956  lgsquadlem2  25957  relsubgr  27051  erclwwlkrel  27795  erclwwlknrel  27845  vcrel  28337  h2hlm  28757  hlimi  28965  relmntop  31265  relae  31499  dmscut  33272  fnerel  33686  filnetlem3  33728  brabg2  35006  heiborlem3  35106  heiborlem4  35107  relrngo  35189  isdivrngo  35243  drngoi  35244  isdrngo1  35249  riscer  35281  relcoss  35683  relssr  35755  prter1  36030  prter3  36033  prjsper  39278  reldvds  40667  nelbrim  43494  isomgrrel  44007  rellininds  44518