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Theorem ot2ndg 6302
Description: Extract the second member of an ordered triple. (See ot1stg 6301 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 3768 . . . . . 6  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
21fveq2i 5672 . . . . 5  |-  ( 1st `  <. A ,  B ,  C >. )  =  ( 1st `  <. <. A ,  B >. ,  C >. )
3 opex 4369 . . . . . 6  |-  <. A ,  B >.  e.  _V
4 op1stg 6299 . . . . . 6  |-  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  X )  ->  ( 1st `  <. <. A ,  B >. ,  C >. )  =  <. A ,  B >. )
53, 4mpan 652 . . . . 5  |-  ( C  e.  X  ->  ( 1st `  <. <. A ,  B >. ,  C >. )  =  <. A ,  B >. )
62, 5syl5eq 2432 . . . 4  |-  ( C  e.  X  ->  ( 1st `  <. A ,  B ,  C >. )  =  <. A ,  B >. )
76fveq2d 5673 . . 3  |-  ( C  e.  X  ->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  ( 2nd `  <. A ,  B >. ) )
8 op2ndg 6300 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
97, 8sylan9eqr 2442 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  C  e.  X )  ->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  B )
1093impa 1148 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 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2900   <.cop 3761   <.cotp 3762   ` cfv 5395   1stc1st 6287   2ndc2nd 6288
This theorem is referenced by:  splval  11708  mamufval  27113  mapdhval  31840  hdmap1val  31915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-ot 3768  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-iota 5359  df-fun 5397  df-fv 5403  df-1st 6289  df-2nd 6290
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