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Theorem ot2ndg 6694
Description: Extract the second member of an ordered triple. (See ot1stg 6693 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 3986 . . . . . 6  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
21fveq2i 5794 . . . . 5  |-  ( 1st `  <. A ,  B ,  C >. )  =  ( 1st `  <. <. A ,  B >. ,  C >. )
3 opex 4656 . . . . . 6  |-  <. A ,  B >.  e.  _V
4 op1stg 6691 . . . . . 6  |-  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  X )  ->  ( 1st `  <. <. A ,  B >. ,  C >. )  =  <. A ,  B >. )
53, 4mpan 670 . . . . 5  |-  ( C  e.  X  ->  ( 1st `  <. <. A ,  B >. ,  C >. )  =  <. A ,  B >. )
62, 5syl5eq 2504 . . . 4  |-  ( C  e.  X  ->  ( 1st `  <. A ,  B ,  C >. )  =  <. A ,  B >. )
76fveq2d 5795 . . 3  |-  ( C  e.  X  ->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  ( 2nd `  <. A ,  B >. ) )
8 op2ndg 6692 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
97, 8sylan9eqr 2514 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  C  e.  X )  ->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  B )
1093impa 1183 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3070   <.cop 3983   <.cotp 3985   ` cfv 5518   1stc1st 6677   2ndc2nd 6678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-ot 3986  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-iota 5481  df-fun 5520  df-fv 5526  df-1st 6679  df-2nd 6680
This theorem is referenced by:  splval  12497  mamufval  18394  el2xptp0  30267  oteqimp  30269  el2spthonot0  30530  usg2spot2nb  30798  usgreg2spot  30800  2spotmdisj  30801  mapdhval  35677  hdmap1val  35752
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