Theorem op2ndb 6130

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 5386 to extract the first member, op2nda 6131 for an alternate version, and op2nd 7840 for the preferred version.) (Contributed by NM, 25-Nov-2003.)

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 6129	. . . . . 6 ⊢ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
4	3	inteqi 4883	. . . . 5 ⊢ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ {⟨𝐵, 𝐴⟩}
5		opex 5379	. . . . . 6 ⊢ ⟨𝐵, 𝐴⟩ ∈ V
6	5	intsn 4917	. . . . 5 ⊢ ∩ {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴⟩
7	4, 6	eqtri 2766	. . . 4 ⊢ ∩ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩
8	7	inteqi 4883	. . 3 ⊢ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ ⟨𝐵, 𝐴⟩
9	8	inteqi 4883	. 2 ⊢ ∩ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ ∩ ⟨𝐵, 𝐴⟩
10	2, 1	op1stb 5386	. 2 ⊢ ∩ ∩ ⟨𝐵, 𝐴⟩ = 𝐵
11	9, 10	eqtri 2766	1 ⊢ ∩ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = 𝐵

Syntax hints:   = wceq 1539   ∈ wcel 2106  Vcvv 3432  {csn 4561  ⟨cop 4567  ∩ cint 4879  ^◡ccnv 5588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-int 4880  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597

This theorem is referenced by:  2ndval2  7849