Theorem op2ndb 5211

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4568 to extract the first member and op2nda 5212 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 5210	. . . . . 6 |- `' { <. A , B >. } = { <. B , A >. }
4	3	inteqi 3926	. . . . 5 |- |^| `' { <. A , B >. } = |^| { <. B , A >. }
5	2, 1	opex 4314	. . . . . 6 |- <. B , A >. e. _V
6	5	intsn 3957	. . . . 5 |- |^| { <. B , A >. } = <. B , A >.
7	4, 6	eqtri 2250	. . . 4 |- |^| `' { <. A , B >. } = <. B , A >.
8	7	inteqi 3926	. . 3 |- |^| |^| `' { <. A , B >. } = |^| <. B , A >.
9	8	inteqi 3926	. 2 |- |^| |^| |^| `' { <. A , B >. } = |^| |^| <. B , A >.
10	2, 1	op1stb 4568	. 2 |- |^| |^| <. B , A >. = B
11	9, 10	eqtri 2250	1 |- |^| |^| |^| `' { <. A , B >. } = B

Syntax hints:   = wceq 1395   e. wcel 2200   _Vcvv 2799   {csn 3666   <.cop 3669   |^|cint 3922   `'ccnv 4717

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-int 3923  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-cnv 4726

This theorem is referenced by:  2ndval2  6300