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Theorem op2ndb 4901
Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4290 to extract the first member and op2nda 4902 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
StepHypRef Expression
1 cnvsn.1 . . . . . . 7  |-  A  e. 
_V
2 cnvsn.2 . . . . . . 7  |-  B  e. 
_V
31, 2cnvsn 4900 . . . . . 6  |-  `' { <. A ,  B >. }  =  { <. B ,  A >. }
43inteqi 3687 . . . . 5  |-  |^| `' { <. A ,  B >. }  =  |^| { <. B ,  A >. }
52, 1opex 4047 . . . . . 6  |-  <. B ,  A >.  e.  _V
65intsn 3718 . . . . 5  |-  |^| { <. B ,  A >. }  =  <. B ,  A >.
74, 6eqtri 2108 . . . 4  |-  |^| `' { <. A ,  B >. }  =  <. B ,  A >.
87inteqi 3687 . . 3  |-  |^| |^| `' { <. A ,  B >. }  =  |^| <. B ,  A >.
98inteqi 3687 . 2  |-  |^| |^| |^| `' { <. A ,  B >. }  =  |^| |^| <. B ,  A >.
102, 1op1stb 4290 . 2  |-  |^| |^| <. B ,  A >.  =  B
119, 10eqtri 2108 1  |-  |^| |^| |^| `' { <. A ,  B >. }  =  B
Colors of variables: wff set class
Syntax hints:    = wceq 1289    e. wcel 1438   _Vcvv 2619   {csn 3441   <.cop 3444   |^|cint 3683   `'ccnv 4427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-int 3684  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436
This theorem is referenced by:  2ndval2  5909
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