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Theorem ot3rdg 6823
Description: Extract the third member of an ordered triple. (See ot1stg 6821 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 4011 . . 3  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
21fveq2i 5884 . 2  |-  ( 2nd `  <. A ,  B ,  C >. )  =  ( 2nd `  <. <. A ,  B >. ,  C >. )
3 opex 4686 . . 3  |-  <. A ,  B >.  e.  _V
4 op2ndg 6820 . . 3  |-  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  V )  ->  ( 2nd `  <. <. A ,  B >. ,  C >. )  =  C )
53, 4mpan 674 . 2  |-  ( C  e.  V  ->  ( 2nd `  <. <. A ,  B >. ,  C >. )  =  C )
62, 5syl5eq 2482 1  |-  ( C  e.  V  ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   _Vcvv 3087   <.cop 4008   <.cotp 4010   ` cfv 5601   2ndc2nd 6806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-ot 4011  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fv 5609  df-2nd 6808
This theorem is referenced by:  oteqimp  6826  el2xptp0  6851  splval  12843  splcl  12844  ida2  15905  coa2  15915  mamufval  19341  frg2woteq  25633  msrval  29964  mapdhval  35000  hdmap1val  35075
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