Theorem op2ndb 5149

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4509 to extract the first member and op2nda 5150 for an alternate version.) (Contributed by NM, 25-Nov-2003.)

Hypotheses

Ref	Expression
cnvsn.1	|- A e. _V
cnvsn.2	|- B e. _V

Assertion

Ref	Expression
op2ndb	|- |^| |^| |^| `' { <. A , B >. } = B

Proof of Theorem op2ndb

Step	Hyp	Ref	Expression
1		cnvsn.1	. . . . . . 7 |- A e. _V
2		cnvsn.2	. . . . . . 7 |- B e. _V
3	1, 2	cnvsn 5148	. . . . . 6 |- `' { <. A , B >. } = { <. B , A >. }
4	3	inteqi 3874	. . . . 5 |- |^| `' { <. A , B >. } = |^| { <. B , A >. }
5	2, 1	opex 4258	. . . . . 6 |- <. B , A >. e. _V
6	5	intsn 3905	. . . . 5 |- |^| { <. B , A >. } = <. B , A >.
7	4, 6	eqtri 2214	. . . 4 |- |^| `' { <. A , B >. } = <. B , A >.
8	7	inteqi 3874	. . 3 |- |^| |^| `' { <. A , B >. } = |^| <. B , A >.
9	8	inteqi 3874	. 2 |- |^| |^| |^| `' { <. A , B >. } = |^| |^| <. B , A >.
10	2, 1	op1stb 4509	. 2 |- |^| |^| <. B , A >. = B
11	9, 10	eqtri 2214	1 |- |^| |^| |^| `' { <. A , B >. } = B

Syntax hints:   = wceq 1364   e. wcel 2164   _Vcvv 2760   {csn 3618   <.cop 3621   |^|cint 3870   `'ccnv 4658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-int 3871  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667

This theorem is referenced by:  2ndval2  6209