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Theorem op2ndb 5154
Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4568 to extract the first member, op2nda 5155 for an alternate version, and op2nd 6091 for the preferred 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
StepHypRef Expression
1 cnvsn.1 . . . . . . 7  |-  A  e. 
_V
2 cnvsn.2 . . . . . . 7  |-  B  e. 
_V
31, 2cnvsn 5153 . . . . . 6  |-  `' { <. A ,  B >. }  =  { <. B ,  A >. }
43inteqi 3867 . . . . 5  |-  |^| `' { <. A ,  B >. }  =  |^| { <. B ,  A >. }
5 opex 4236 . . . . . 6  |-  <. B ,  A >.  e.  _V
65intsn 3899 . . . . 5  |-  |^| { <. B ,  A >. }  =  <. B ,  A >.
74, 6eqtri 2304 . . . 4  |-  |^| `' { <. A ,  B >. }  =  <. B ,  A >.
87inteqi 3867 . . 3  |-  |^| |^| `' { <. A ,  B >. }  =  |^| <. B ,  A >.
98inteqi 3867 . 2  |-  |^| |^| |^| `' { <. A ,  B >. }  =  |^| |^| <. B ,  A >.
102, 1op1stb 4568 . 2  |-  |^| |^| <. B ,  A >.  =  B
119, 10eqtri 2304 1  |-  |^| |^| |^| `' { <. A ,  B >. }  =  B
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1685   _Vcvv 2789   {csn 3641   <.cop 3644   |^|cint 3863   `'ccnv 4687
This theorem is referenced by:  2ndval2  6100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-int 3864  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696
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