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Theorem ot3rdg 6304
Description: Extract the third member of an ordered triple. (See ot1stg 6302 comment.) (Contributed by NM, 3-Apr-2015.)
Assertion
Ref Expression
ot3rdg  |-  ( C  e.  V  ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )

Proof of Theorem ot3rdg
StepHypRef Expression
1 df-ot 3769 . . 3  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
21fveq2i 5673 . 2  |-  ( 2nd `  <. A ,  B ,  C >. )  =  ( 2nd `  <. <. A ,  B >. ,  C >. )
3 opex 4370 . . 3  |-  <. A ,  B >.  e.  _V
4 op2ndg 6301 . . 3  |-  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  V )  ->  ( 2nd `  <. <. A ,  B >. ,  C >. )  =  C )
53, 4mpan 652 . 2  |-  ( C  e.  V  ->  ( 2nd `  <. <. A ,  B >. ,  C >. )  =  C )
62, 5syl5eq 2433 1  |-  ( C  e.  V  ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2901   <.cop 3762   <.cotp 3763   ` cfv 5396   2ndc2nd 6289
This theorem is referenced by:  splval  11709  splcl  11710  ida2  14143  coa2  14153  mamufval  27114  mapdhval  31841  hdmap1val  31916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-ot 3769  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-iota 5360  df-fun 5398  df-fv 5404  df-2nd 6291
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