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Theorem op2ndb 5344
Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4749 to extract the first member, op2nda 5345 for an alternate version, and op2nd 6347 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 5343 . . . . . 6  |-  `' { <. A ,  B >. }  =  { <. B ,  A >. }
43inteqi 4046 . . . . 5  |-  |^| `' { <. A ,  B >. }  =  |^| { <. B ,  A >. }
5 opex 4419 . . . . . 6  |-  <. B ,  A >.  e.  _V
65intsn 4078 . . . . 5  |-  |^| { <. B ,  A >. }  =  <. B ,  A >.
74, 6eqtri 2455 . . . 4  |-  |^| `' { <. A ,  B >. }  =  <. B ,  A >.
87inteqi 4046 . . 3  |-  |^| |^| `' { <. A ,  B >. }  =  |^| <. B ,  A >.
98inteqi 4046 . 2  |-  |^| |^| |^| `' { <. A ,  B >. }  =  |^| |^| <. B ,  A >.
102, 1op1stb 4749 . 2  |-  |^| |^| <. B ,  A >.  =  B
119, 10eqtri 2455 1  |-  |^| |^| |^| `' { <. A ,  B >. }  =  B
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806   <.cop 3809   |^|cint 4042   `'ccnv 4868
This theorem is referenced by:  2ndval2  6356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-int 4043  df-br 4205  df-opab 4259  df-xp 4875  df-rel 4876  df-cnv 4877
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