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Theorem ot3rdgg 6179
Description: Extract the third member of an ordered triple. (See ot1stg 6177 comment.) (Contributed by NM, 3-Apr-2015.)
Assertion
Ref Expression
ot3rdgg  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )

Proof of Theorem ot3rdgg
StepHypRef Expression
1 df-ot 3617 . . 3  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
21fveq2i 5537 . 2  |-  ( 2nd `  <. A ,  B ,  C >. )  =  ( 2nd `  <. <. A ,  B >. ,  C >. )
3 opexg 4246 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  e. 
_V )
4 op2ndg 6176 . . . 4  |-  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  X )  ->  ( 2nd `  <. <. A ,  B >. ,  C >. )  =  C )
53, 4sylan 283 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  C  e.  X )  ->  ( 2nd `  <. <. A ,  B >. ,  C >. )  =  C )
653impa 1196 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 2nd `  <. <. A ,  B >. ,  C >. )  =  C )
72, 6eqtrid 2234 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980    = wceq 1364    e. wcel 2160   _Vcvv 2752   <.cop 3610   <.cotp 3611   ` cfv 5235   2ndc2nd 6164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-ot 3617  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fv 5243  df-2nd 6166
This theorem is referenced by: (None)
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