Theorem op2ndb 5227

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4581 to extract the first member and op2nda 5228 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 5226	. . . . . 6 |- `' { <. A , B >. } = { <. B , A >. }
4	3	inteqi 3937	. . . . 5 |- |^| `' { <. A , B >. } = |^| { <. B , A >. }
5	2, 1	opex 4327	. . . . . 6 |- <. B , A >. e. _V
6	5	intsn 3968	. . . . 5 |- |^| { <. B , A >. } = <. B , A >.
7	4, 6	eqtri 2252	. . . 4 |- |^| `' { <. A , B >. } = <. B , A >.
8	7	inteqi 3937	. . 3 |- |^| |^| `' { <. A , B >. } = |^| <. B , A >.
9	8	inteqi 3937	. 2 |- |^| |^| |^| `' { <. A , B >. } = |^| |^| <. B , A >.
10	2, 1	op1stb 4581	. 2 |- |^| |^| <. B , A >. = B
11	9, 10	eqtri 2252	1 |- |^| |^| |^| `' { <. A , B >. } = B

Syntax hints:   = wceq 1398   e. wcel 2202   _Vcvv 2803   {csn 3673   <.cop 3676   |^|cint 3933   `'ccnv 4730

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-int 3934  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739

This theorem is referenced by:  2ndval2  6328