Theorem op2ndb 5153

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4513 to extract the first member and op2nda 5154 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 5152	. . . . . 6 |- `' { <. A , B >. } = { <. B , A >. }
4	3	inteqi 3878	. . . . 5 |- |^| `' { <. A , B >. } = |^| { <. B , A >. }
5	2, 1	opex 4262	. . . . . 6 |- <. B , A >. e. _V
6	5	intsn 3909	. . . . 5 |- |^| { <. B , A >. } = <. B , A >.
7	4, 6	eqtri 2217	. . . 4 |- |^| `' { <. A , B >. } = <. B , A >.
8	7	inteqi 3878	. . 3 |- |^| |^| `' { <. A , B >. } = |^| <. B , A >.
9	8	inteqi 3878	. 2 |- |^| |^| |^| `' { <. A , B >. } = |^| |^| <. B , A >.
10	2, 1	op1stb 4513	. 2 |- |^| |^| <. B , A >. = B
11	9, 10	eqtri 2217	1 |- |^| |^| |^| `' { <. A , B >. } = B

Syntax hints:   = wceq 1364   e. wcel 2167   _Vcvv 2763   {csn 3622   <.cop 3625   |^|cint 3874   `'ccnv 4662

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-int 3875  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671

This theorem is referenced by:  2ndval2  6214