Theorem op2ndb 5266

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4619 to extract the first member and op2nda 5267 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 5265	. . . . . 6 |- `' { <. A , B >. } = { <. B , A >. }
4	3	inteqi 3969	. . . . 5 |- |^| `' { <. A , B >. } = |^| { <. B , A >. }
5	2, 1	opex 4364	. . . . . 6 |- <. B , A >. e. _V
6	5	intsn 4000	. . . . 5 |- |^| { <. B , A >. } = <. B , A >.
7	4, 6	eqtri 2259	. . . 4 |- |^| `' { <. A , B >. } = <. B , A >.
8	7	inteqi 3969	. . 3 |- |^| |^| `' { <. A , B >. } = |^| <. B , A >.
9	8	inteqi 3969	. 2 |- |^| |^| |^| `' { <. A , B >. } = |^| |^| <. B , A >.
10	2, 1	op1stb 4619	. 2 |- |^| |^| <. B , A >. = B
11	9, 10	eqtri 2259	1 |- |^| |^| |^| `' { <. A , B >. } = B

Syntax hints:   = wceq 1402   e. wcel 2209   _Vcvv 2821   {csn 3705   <.cop 3708   |^|cint 3965   `'ccnv 4768

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-int 3966  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777

This theorem is referenced by:  2ndval2  6380