Theorem op2ndb 5220

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4575 to extract the first member and op2nda 5221 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 5219	. . . . . 6 |- `' { <. A , B >. } = { <. B , A >. }
4	3	inteqi 3932	. . . . 5 |- |^| `' { <. A , B >. } = |^| { <. B , A >. }
5	2, 1	opex 4321	. . . . . 6 |- <. B , A >. e. _V
6	5	intsn 3963	. . . . 5 |- |^| { <. B , A >. } = <. B , A >.
7	4, 6	eqtri 2252	. . . 4 |- |^| `' { <. A , B >. } = <. B , A >.
8	7	inteqi 3932	. . 3 |- |^| |^| `' { <. A , B >. } = |^| <. B , A >.
9	8	inteqi 3932	. 2 |- |^| |^| |^| `' { <. A , B >. } = |^| |^| <. B , A >.
10	2, 1	op1stb 4575	. 2 |- |^| |^| <. B , A >. = B
11	9, 10	eqtri 2252	1 |- |^| |^| |^| `' { <. A , B >. } = B

Syntax hints:   = wceq 1397   e. wcel 2202   _Vcvv 2802   {csn 3669   <.cop 3672   |^|cint 3928   `'ccnv 4724

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-int 3929  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733

This theorem is referenced by:  2ndval2  6318