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Theorem ot3rdg 6810
Description: Extract the third member of an ordered triple. (See ot1stg 6808 comment.) (Contributed by NM, 3-Apr-2015.)
Assertion
Ref Expression
ot3rdg  |-  ( C  e.  V  ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )

Proof of Theorem ot3rdg
StepHypRef Expression
1 df-ot 4041 . . 3  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
21fveq2i 5874 . 2  |-  ( 2nd `  <. A ,  B ,  C >. )  =  ( 2nd `  <. <. A ,  B >. ,  C >. )
3 opex 4716 . . 3  |-  <. A ,  B >.  e.  _V
4 op2ndg 6807 . . 3  |-  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  V )  ->  ( 2nd `  <. <. A ,  B >. ,  C >. )  =  C )
53, 4mpan 670 . 2  |-  ( C  e.  V  ->  ( 2nd `  <. <. A ,  B >. ,  C >. )  =  C )
62, 5syl5eq 2520 1  |-  ( C  e.  V  ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3118   <.cop 4038   <.cotp 4040   ` cfv 5593   2ndc2nd 6793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-ot 4041  df-uni 4251  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fv 5601  df-2nd 6795
This theorem is referenced by:  oteqimp  6813  el2xptp0  6838  splval  12702  splcl  12703  ida2  15256  coa2  15266  mamufval  18733  frg2woteq  24852  msrval  28691  mapdhval  36814  hdmap1val  36889
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