Theorem relopab 2494

Description: A class of ordered pairs is a relation.

Assertion

Ref	Expression
relopab	⊢ Rel {⟨x, y⟩∣φ}

Distinct variable group(s):   x,y

Proof of Theorem relopab

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . 6 ⊢ x ∈ V
2		visset 1350	. . . . . 6 ⊢ y ∈ V
3	1, 2	pm3.2i 234	. . . . 5 ⊢ (x ∈ V ∧ y ∈ V)
4	3	a1i 7	. . . 4 ⊢ (φ → (x ∈ V ∧ y ∈ V))
5	4	ssopab2i 2120	. . 3 ⊢ {⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣(x ∈ V ∧ y ∈ V)}
6		df-xp 2424	. . 3 ⊢ (V × V) = {⟨x, y⟩∣(x ∈ V ∧ y ∈ V)}
7	5, 6	sseqtr4 1533	. 2 ⊢ {⟨x, y⟩∣φ} ⊆ (V × V)
8		df-rel 2425	. 2 ⊢ (Rel {⟨x, y⟩∣φ} ↔ {⟨x, y⟩∣φ} ⊆ (V × V))
9	7, 8	mpbir 165	1 ⊢ Rel {⟨x, y⟩∣φ}

Syntax hints:   ∧ wa 196   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  {copab 2055   × cxp 2408  Rel wrel 2415

This theorem is referenced by:  inopab 2495  reli 2500  rele 2501  relcnv 2624  cnvopab 2632  relco 2658  funopab 2694  fnopabfv 2858  reloprab 3022  reldmoprab 3034  relen 3277  reldom 3278  aceq3lem 3555  aceq3 3556

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-rel 2425

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