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Theorem ot2ndg 6354
Description: Extract the second member of an ordered triple. (See ot1stg 6353 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 3816 . . . . . 6  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
21fveq2i 5723 . . . . 5  |-  ( 1st `  <. A ,  B ,  C >. )  =  ( 1st `  <. <. A ,  B >. ,  C >. )
3 opex 4419 . . . . . 6  |-  <. A ,  B >.  e.  _V
4 op1stg 6351 . . . . . 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 2479 . . . 4  |-  ( C  e.  X  ->  ( 1st `  <. A ,  B ,  C >. )  =  <. A ,  B >. )
76fveq2d 5724 . . 3  |-  ( C  e.  X  ->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  ( 2nd `  <. A ,  B >. ) )
8 op2ndg 6352 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
97, 8sylan9eqr 2489 . 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 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   <.cotp 3810   ` cfv 5446   1stc1st 6339   2ndc2nd 6340
This theorem is referenced by:  splval  11772  mamufval  27411  el2xptp0  28051  oteqimp  28053  el2spthonot0  28291  usg2spot2nb  28391  usgreg2spot  28393  2spotmdisj  28394  mapdhval  32459  hdmap1val  32534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-ot 3816  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-1st 6341  df-2nd 6342
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