Theorem op2ndb 6200

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 5431 to extract the first member, op2nda 6201 for an alternate version, and op2nd 7977 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 6199	. . . . . 6 ⊢ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
4	3	inteqi 4914	. . . . 5 ⊢ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ {⟨𝐵, 𝐴⟩}
5		opex 5424	. . . . . 6 ⊢ ⟨𝐵, 𝐴⟩ ∈ V
6	5	intsn 4948	. . . . 5 ⊢ ∩ {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴⟩
7	4, 6	eqtri 2752	. . . 4 ⊢ ∩ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩
8	7	inteqi 4914	. . 3 ⊢ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ ⟨𝐵, 𝐴⟩
9	8	inteqi 4914	. 2 ⊢ ∩ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ ∩ ⟨𝐵, 𝐴⟩
10	2, 1	op1stb 5431	. 2 ⊢ ∩ ∩ ⟨𝐵, 𝐴⟩ = 𝐵
11	9, 10	eqtri 2752	1 ⊢ ∩ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = 𝐵

Syntax hints:   = wceq 1540   ∈ wcel 2109  Vcvv 3447  {csn 4589  ⟨cop 4595  ∩ cint 4910  ^◡ccnv 5637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-int 4911  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646

This theorem is referenced by:  2ndval2  7986