Theorem op2ndb 5180

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4538 to extract the first member and op2nda 5181 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 5179	. . . . . 6 |- `' { <. A , B >. } = { <. B , A >. }
4	3	inteqi 3898	. . . . 5 |- |^| `' { <. A , B >. } = |^| { <. B , A >. }
5	2, 1	opex 4286	. . . . . 6 |- <. B , A >. e. _V
6	5	intsn 3929	. . . . 5 |- |^| { <. B , A >. } = <. B , A >.
7	4, 6	eqtri 2227	. . . 4 |- |^| `' { <. A , B >. } = <. B , A >.
8	7	inteqi 3898	. . 3 |- |^| |^| `' { <. A , B >. } = |^| <. B , A >.
9	8	inteqi 3898	. 2 |- |^| |^| |^| `' { <. A , B >. } = |^| |^| <. B , A >.
10	2, 1	op1stb 4538	. 2 |- |^| |^| <. B , A >. = B
11	9, 10	eqtri 2227	1 |- |^| |^| |^| `' { <. A , B >. } = B

Syntax hints:   = wceq 1373   e. wcel 2177   _Vcvv 2773   {csn 3638   <.cop 3641   |^|cint 3894   `'ccnv 4687

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-int 3895  df-br 4055  df-opab 4117  df-xp 4694  df-rel 4695  df-cnv 4696

This theorem is referenced by:  2ndval2  6260