Theorem op2ndb 6083

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 5362 to extract the first member, op2nda 6084 for an alternate version, and op2nd 7697 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 6082	. . . . . 6 ⊢ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
4	3	inteqi 4879	. . . . 5 ⊢ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ {⟨𝐵, 𝐴⟩}
5		opex 5355	. . . . . 6 ⊢ ⟨𝐵, 𝐴⟩ ∈ V
6	5	intsn 4911	. . . . 5 ⊢ ∩ {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴⟩
7	4, 6	eqtri 2844	. . . 4 ⊢ ∩ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩
8	7	inteqi 4879	. . 3 ⊢ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ ⟨𝐵, 𝐴⟩
9	8	inteqi 4879	. 2 ⊢ ∩ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ ∩ ⟨𝐵, 𝐴⟩
10	2, 1	op1stb 5362	. 2 ⊢ ∩ ∩ ⟨𝐵, 𝐴⟩ = 𝐵
11	9, 10	eqtri 2844	1 ⊢ ∩ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = 𝐵

Syntax hints:   = wceq 1533   ∈ wcel 2110  Vcvv 3494  {csn 4566  ⟨cop 4572  ∩ cint 4875  ^◡ccnv 5553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-int 4876  df-br 5066  df-opab 5128  df-xp 5560  df-rel 5561  df-cnv 5562

This theorem is referenced by:  2ndval2  7706