Theorem op2ndb 6231

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 5473 to extract the first member, op2nda 6232 for an alternate version, and op2nd 8002 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 6230	. . . . . 6 ⊢ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
4	3	inteqi 4953	. . . . 5 ⊢ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ {⟨𝐵, 𝐴⟩}
5		opex 5466	. . . . . 6 ⊢ ⟨𝐵, 𝐴⟩ ∈ V
6	5	intsn 4989	. . . . 5 ⊢ ∩ {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴⟩
7	4, 6	eqtri 2756	. . . 4 ⊢ ∩ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩
8	7	inteqi 4953	. . 3 ⊢ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ ⟨𝐵, 𝐴⟩
9	8	inteqi 4953	. 2 ⊢ ∩ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ ∩ ⟨𝐵, 𝐴⟩
10	2, 1	op1stb 5473	. 2 ⊢ ∩ ∩ ⟨𝐵, 𝐴⟩ = 𝐵
11	9, 10	eqtri 2756	1 ⊢ ∩ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = 𝐵

Syntax hints:   = wceq 1534   ∈ wcel 2099  Vcvv 3471  {csn 4629  ⟨cop 4635  ∩ cint 4949  ^◡ccnv 5677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-int 4950  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686

This theorem is referenced by:  2ndval2  8011