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Theorem op2ndb 5108
Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4506 to extract the first member, op2nda 5109 for an alternate version, and op2nd 6028 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 5107 . . . . . 6  |-  `' { <. A ,  B >. }  =  { <. B ,  A >. }
43inteqi 3807 . . . . 5  |-  |^| `' { <. A ,  B >. }  =  |^| { <. B ,  A >. }
5 opex 4174 . . . . . 6  |-  <. B ,  A >.  e.  _V
65intsn 3839 . . . . 5  |-  |^| { <. B ,  A >. }  =  <. B ,  A >.
74, 6eqtri 2276 . . . 4  |-  |^| `' { <. A ,  B >. }  =  <. B ,  A >.
87inteqi 3807 . . 3  |-  |^| |^| `' { <. A ,  B >. }  =  |^| <. B ,  A >.
98inteqi 3807 . 2  |-  |^| |^| |^| `' { <. A ,  B >. }  =  |^| |^| <. B ,  A >.
102, 1op1stb 4506 . 2  |-  |^| |^| <. B ,  A >.  =  B
119, 10eqtri 2276 1  |-  |^| |^| |^| `' { <. A ,  B >. }  =  B
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. wcel 1621   _Vcvv 2740   {csn 3581   <.cop 3584   |^|cint 3803   `'ccnv 4625
This theorem is referenced by:  2ndval2  6037
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-int 3804  df-br 3964  df-opab 4018  df-xp 4640  df-rel 4641  df-cnv 4642
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