Theorem relopabi 4737

Description: A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)

Hypothesis

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

Assertion

Ref	Expression
relopabi	⊢ Rel 𝐴

Proof of Theorem relopabi

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopabi.1	. . . 4 ⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		df-opab 4051	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
3	1, 2	eqtri 2191	. . 3 ⊢ 𝐴 = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
4		vex 2733	. . . . . . . 8 ⊢ 𝑥 ∈ V
5		vex 2733	. . . . . . . 8 ⊢ 𝑦 ∈ V
6	4, 5	opelvv 4661	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ (V × V)
7		eleq1 2233	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (V × V) ↔ ⟨𝑥, 𝑦⟩ ∈ (V × V)))
8	6, 7	mpbiri 167	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 ∈ (V × V))
9	8	adantr 274	. . . . 5 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (V × V))
10	9	exlimivv 1889	. . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (V × V))
11	10	abssi 3222	. . 3 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ (V × V)
12	3, 11	eqsstri 3179	. 2 ⊢ 𝐴 ⊆ (V × V)
13		df-rel 4618	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
14	12, 13	mpbir 145	1 ⊢ Rel 𝐴

Syntax hints:   ∧ wa 103   = wceq 1348  ∃wex 1485   ∈ wcel 2141  {cab 2156  Vcvv 2730   ⊆ wss 3121  ⟨cop 3586  {copab 4049   × cxp 4609  Rel wrel 4616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-xp 4617  df-rel 4618

This theorem is referenced by:  relopab  4738  mptrel  4739  reli  4740  rele  4741  relcnv  4989  cotr  4992  relco  5109  reloprab  5901  reldmoprab  5938  eqer  6545  ecopover  6611  ecopoverg  6614  relen  6722  reldom  6723  enq0er  7397  aprcl  8565  climrel  11243  brstruct  12425