Theorem opabss 2100

Description: The collection of ordered pairs in a class is a subclass of it.

Assertion

Ref	Expression
opabss	⊢ {⟨x, y⟩∣xRy} ⊆ R

Distinct variable group(s):   x,R   y,R

Proof of Theorem opabss

Step	Hyp	Ref	Expression
1		df-opab 2098	. . 3 ⊢ {⟨x, y⟩∣xRy} = {z∣∃x∃y(z = ⟨x, y⟩ ∧ xRy)}
2		eleq1 1149	. . . . . . 7 ⊢ (z = ⟨x, y⟩ → (z ∈ R ↔ ⟨x, y⟩ ∈ R))
3	2	biimpar 325	. . . . . 6 ⊢ ((z = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ R) → z ∈ R)
4		df-br 2063	. . . . . 6 ⊢ (xRy ↔ ⟨x, y⟩ ∈ R)
5	3, 4	sylan2b 347	. . . . 5 ⊢ ((z = ⟨x, y⟩ ∧ xRy) → z ∈ R)
6	5	19.23aivv 953	. . . 4 ⊢ (∃x∃y(z = ⟨x, y⟩ ∧ xRy) → z ∈ R)
7	6	ss2abi 1552	. . 3 ⊢ {z∣∃x∃y(z = ⟨x, y⟩ ∧ xRy)} ⊆ {z∣z ∈ R}
8	1, 7	eqsstr 1530	. 2 ⊢ {⟨x, y⟩∣xRy} ⊆ {z∣z ∈ R}
9		abid2 1186	. 2 ⊢ {z∣z ∈ R} = R
10	8, 9	sseqtr 1532	1 ⊢ {⟨x, y⟩∣xRy} ⊆ R

Syntax hints:   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  ⟨cop 1810   class class class wbr 2054  {copab 2055

This theorem is referenced by:  cotr 2625  cnvsym 2626

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-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492  df-br 2063  df-opab 2098

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