ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ot3rdgg Unicode version

Theorem ot3rdgg 5939
Description: Extract the third member of an ordered triple. (See ot1stg 5937 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 3460 . . 3  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
21fveq2i 5321 . 2  |-  ( 2nd `  <. A ,  B ,  C >. )  =  ( 2nd `  <. <. A ,  B >. ,  C >. )
3 opexg 4064 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  e. 
_V )
4 op2ndg 5936 . . . 4  |-  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  X )  ->  ( 2nd `  <. <. A ,  B >. ,  C >. )  =  C )
53, 4sylan 278 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  C  e.  X )  ->  ( 2nd `  <. <. A ,  B >. ,  C >. )  =  C )
653impa 1139 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 2nd `  <. <. A ,  B >. ,  C >. )  =  C )
72, 6syl5eq 2133 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 103    /\ w3a 925    = wceq 1290    e. wcel 1439   _Vcvv 2620   <.cop 3453   <.cotp 3454   ` cfv 5028   2ndc2nd 5924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-ot 3460  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-iota 4993  df-fun 5030  df-fv 5036  df-2nd 5926
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator