MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  op2ndb Unicode version

Theorem op2ndb 5143
Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4541 to extract the first member, op2nda 5144 for an alternate version, and op2nd 6063 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 5142 . . . . . 6  |-  `' { <. A ,  B >. }  =  { <. B ,  A >. }
43inteqi 3840 . . . . 5  |-  |^| `' { <. A ,  B >. }  =  |^| { <. B ,  A >. }
5 opex 4209 . . . . . 6  |-  <. B ,  A >.  e.  _V
65intsn 3872 . . . . 5  |-  |^| { <. B ,  A >. }  =  <. B ,  A >.
74, 6eqtri 2278 . . . 4  |-  |^| `' { <. A ,  B >. }  =  <. B ,  A >.
87inteqi 3840 . . 3  |-  |^| |^| `' { <. A ,  B >. }  =  |^| <. B ,  A >.
98inteqi 3840 . 2  |-  |^| |^| |^| `' { <. A ,  B >. }  =  |^| |^| <. B ,  A >.
102, 1op1stb 4541 . 2  |-  |^| |^| <. B ,  A >.  =  B
119, 10eqtri 2278 1  |-  |^| |^| |^| `' { <. A ,  B >. }  =  B
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. wcel 1621   _Vcvv 2763   {csn 3614   <.cop 3617   |^|cint 3836   `'ccnv 4660
This theorem is referenced by:  2ndval2  6072
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 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-int 3837  df-br 3998  df-opab 4052  df-xp 4675  df-rel 4676  df-cnv 4677
  Copyright terms: Public domain W3C validator