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Theorem ot2ndg 6019
Description: Extract the second member of an ordered triple. (See ot1stg 6018 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
Assertion
Ref Expression
ot2ndg  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  B )

Proof of Theorem ot2ndg
StepHypRef Expression
1 df-ot 3507 . . . . 5  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
21fveq2i 5392 . . . 4  |-  ( 1st `  <. A ,  B ,  C >. )  =  ( 1st `  <. <. A ,  B >. ,  C >. )
3 opexg 4120 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  e. 
_V )
4 op1stg 6016 . . . . . 6  |-  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  X )  ->  ( 1st `  <. <. A ,  B >. ,  C >. )  =  <. A ,  B >. )
53, 4sylan 281 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  C  e.  X )  ->  ( 1st `  <. <. A ,  B >. ,  C >. )  =  <. A ,  B >. )
653impa 1161 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 1st `  <. <. A ,  B >. ,  C >. )  =  <. A ,  B >. )
72, 6syl5eq 2162 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 1st `  <. A ,  B ,  C >. )  =  <. A ,  B >. )
87fveq2d 5393 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  ( 2nd `  <. A ,  B >. )
)
9 op2ndg 6017 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
1093adant3 986 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 2nd `  <. A ,  B >. )  =  B )
118, 10eqtrd 2150 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 947    = wceq 1316    e. wcel 1465   _Vcvv 2660   <.cop 3500   <.cotp 3501   ` cfv 5093   1stc1st 6004   2ndc2nd 6005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-ot 3507  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fv 5101  df-1st 6006  df-2nd 6007
This theorem is referenced by: (None)
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