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Theorem ot2ndg 5924
Description: Extract the second member of an ordered triple. (See ot1stg 5923 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 3456 . . . . 5  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
21fveq2i 5308 . . . 4  |-  ( 1st `  <. A ,  B ,  C >. )  =  ( 1st `  <. <. A ,  B >. ,  C >. )
3 opexg 4055 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  e. 
_V )
4 op1stg 5921 . . . . . 6  |-  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  X )  ->  ( 1st `  <. <. A ,  B >. ,  C >. )  =  <. A ,  B >. )
53, 4sylan 277 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  C  e.  X )  ->  ( 1st `  <. <. A ,  B >. ,  C >. )  =  <. A ,  B >. )
653impa 1138 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 1st `  <. <. A ,  B >. ,  C >. )  =  <. A ,  B >. )
72, 6syl5eq 2132 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 1st `  <. A ,  B ,  C >. )  =  <. A ,  B >. )
87fveq2d 5309 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  ( 2nd `  <. A ,  B >. )
)
9 op2ndg 5922 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
1093adant3 963 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 2nd `  <. A ,  B >. )  =  B )
118, 10eqtrd 2120 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 102    /\ w3a 924    = wceq 1289    e. wcel 1438   _Vcvv 2619   <.cop 3449   <.cotp 3450   ` cfv 5015   1stc1st 5909   2ndc2nd 5910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-ot 3456  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fv 5023  df-1st 5911  df-2nd 5912
This theorem is referenced by: (None)
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