Theorem op2ndb 2638

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 1992 to extract the first member and op2nda 2639 for an alternate version.)

Hypotheses

Ref	Expression
cnvsn.1	⊢ A ∈ V
cnvsn.2	⊢ B ∈ V

Assertion

Ref	Expression
op2ndb	⊢ ∩∩∩◡{⟨A, B⟩} = B

Proof of Theorem op2ndb

Step	Hyp	Ref	Expression
1		cnvsn.1	. . . . . . 7 ⊢ A ∈ V
2		cnvsn.2	. . . . . . 7 ⊢ B ∈ V
3	1, 2	cnvsn 2636	. . . . . 6 ⊢ ◡{⟨A, B⟩} = {⟨B, A⟩}
4	3	inteqi 1969	. . . . 5 ⊢ ∩◡{⟨A, B⟩} = ∩{⟨B, A⟩}
5		opex 1893	. . . . . 6 ⊢ ⟨B, A⟩ ∈ V
6	5	intsn 1991	. . . . 5 ⊢ ∩{⟨B, A⟩} = ⟨B, A⟩
7	4, 6	eqtr 1119	. . . 4 ⊢ ∩◡{⟨A, B⟩} = ⟨B, A⟩
8	7	inteqi 1969	. . 3 ⊢ ∩∩◡{⟨A, B⟩} = ∩⟨B, A⟩
9	8	inteqi 1969	. 2 ⊢ ∩∩∩◡{⟨A, B⟩} = ∩∩⟨B, A⟩
10	2	op1stb 1992	. 2 ⊢ ∩∩⟨B, A⟩ = B
11	9, 10	eqtr 1119	1 ⊢ ∩∩∩◡{⟨A, B⟩} = B

Syntax hints:   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {csn 1808  ⟨cop 1810  ∩cint 1965  ^◡ccnv 2409

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-ral 1205  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-int 1966  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426

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