Theorem op2ndb 4980

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4359 to extract the first member and op2nda 4981 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 4979	. . . . . 6 |- `' { <. A , B >. } = { <. B , A >. }
4	3	inteqi 3741	. . . . 5 |- |^| `' { <. A , B >. } = |^| { <. B , A >. }
5	2, 1	opex 4111	. . . . . 6 |- <. B , A >. e. _V
6	5	intsn 3772	. . . . 5 |- |^| { <. B , A >. } = <. B , A >.
7	4, 6	eqtri 2135	. . . 4 |- |^| `' { <. A , B >. } = <. B , A >.
8	7	inteqi 3741	. . 3 |- |^| |^| `' { <. A , B >. } = |^| <. B , A >.
9	8	inteqi 3741	. 2 |- |^| |^| |^| `' { <. A , B >. } = |^| |^| <. B , A >.
10	2, 1	op1stb 4359	. 2 |- |^| |^| <. B , A >. = B
11	9, 10	eqtri 2135	1 |- |^| |^| |^| `' { <. A , B >. } = B

Syntax hints:   = wceq 1314   e. wcel 1463   _Vcvv 2657   {csn 3493   <.cop 3496   |^|cint 3737   `'ccnv 4498

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-int 3738  df-br 3896  df-opab 3950  df-xp 4505  df-rel 4506  df-cnv 4507

This theorem is referenced by:  2ndval2  6008