Theorem op2ndb 5251

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4604 to extract the first member and op2nda 5252 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 5250	. . . . . 6 |- `' { <. A , B >. } = { <. B , A >. }
4	3	inteqi 3958	. . . . 5 |- |^| `' { <. A , B >. } = |^| { <. B , A >. }
5	2, 1	opex 4350	. . . . . 6 |- <. B , A >. e. _V
6	5	intsn 3989	. . . . 5 |- |^| { <. B , A >. } = <. B , A >.
7	4, 6	eqtri 2255	. . . 4 |- |^| `' { <. A , B >. } = <. B , A >.
8	7	inteqi 3958	. . 3 |- |^| |^| `' { <. A , B >. } = |^| <. B , A >.
9	8	inteqi 3958	. 2 |- |^| |^| |^| `' { <. A , B >. } = |^| |^| <. B , A >.
10	2, 1	op1stb 4604	. 2 |- |^| |^| <. B , A >. = B
11	9, 10	eqtri 2255	1 |- |^| |^| |^| `' { <. A , B >. } = B

Syntax hints:   = wceq 1398   e. wcel 2205   _Vcvv 2815   {csn 3694   <.cop 3697   |^|cint 3954   `'ccnv 4753

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-int 3955  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-cnv 4762

This theorem is referenced by:  2ndval2  6363