Theorem op2ndb 5166

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4525 to extract the first member and op2nda 5167 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 5165	. . . . . 6 |- `' { <. A , B >. } = { <. B , A >. }
4	3	inteqi 3889	. . . . 5 |- |^| `' { <. A , B >. } = |^| { <. B , A >. }
5	2, 1	opex 4273	. . . . . 6 |- <. B , A >. e. _V
6	5	intsn 3920	. . . . 5 |- |^| { <. B , A >. } = <. B , A >.
7	4, 6	eqtri 2226	. . . 4 |- |^| `' { <. A , B >. } = <. B , A >.
8	7	inteqi 3889	. . 3 |- |^| |^| `' { <. A , B >. } = |^| <. B , A >.
9	8	inteqi 3889	. 2 |- |^| |^| |^| `' { <. A , B >. } = |^| |^| <. B , A >.
10	2, 1	op1stb 4525	. 2 |- |^| |^| <. B , A >. = B
11	9, 10	eqtri 2226	1 |- |^| |^| |^| `' { <. A , B >. } = B

Syntax hints:   = wceq 1373   e. wcel 2176   _Vcvv 2772   {csn 3633   <.cop 3636   |^|cint 3885   `'ccnv 4674

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-int 3886  df-br 4045  df-opab 4106  df-xp 4681  df-rel 4682  df-cnv 4683

This theorem is referenced by:  2ndval2  6242