Theorem op2ndb 5165

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4524 to extract the first member and op2nda 5166 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 5164	. . . . . 6 |- `' { <. A , B >. } = { <. B , A >. }
4	3	inteqi 3888	. . . . 5 |- |^| `' { <. A , B >. } = |^| { <. B , A >. }
5	2, 1	opex 4272	. . . . . 6 |- <. B , A >. e. _V
6	5	intsn 3919	. . . . 5 |- |^| { <. B , A >. } = <. B , A >.
7	4, 6	eqtri 2225	. . . 4 |- |^| `' { <. A , B >. } = <. B , A >.
8	7	inteqi 3888	. . 3 |- |^| |^| `' { <. A , B >. } = |^| <. B , A >.
9	8	inteqi 3888	. 2 |- |^| |^| |^| `' { <. A , B >. } = |^| |^| <. B , A >.
10	2, 1	op1stb 4524	. 2 |- |^| |^| <. B , A >. = B
11	9, 10	eqtri 2225	1 |- |^| |^| |^| `' { <. A , B >. } = B

Syntax hints:   = wceq 1372   e. wcel 2175   _Vcvv 2771   {csn 3632   <.cop 3635   |^|cint 3884   `'ccnv 4673

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-int 3885  df-br 4044  df-opab 4105  df-xp 4680  df-rel 4681  df-cnv 4682

This theorem is referenced by:  2ndval2  6241