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Theorem ot3rdg 6152
Description: Extract the third member of an ordered triple. (See ot1stg 6150 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 3663 . . 3  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
21fveq2i 5544 . 2  |-  ( 2nd `  <. A ,  B ,  C >. )  =  ( 2nd `  <. <. A ,  B >. ,  C >. )
3 opex 4253 . . 3  |-  <. A ,  B >.  e.  _V
4 op2ndg 6149 . . 3  |-  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  V )  ->  ( 2nd `  <. <. A ,  B >. ,  C >. )  =  C )
53, 4mpan 651 . 2  |-  ( C  e.  V  ->  ( 2nd `  <. <. A ,  B >. ,  C >. )  =  C )
62, 5syl5eq 2340 1  |-  ( C  e.  V  ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   <.cotp 3657   ` cfv 5271   2ndc2nd 6137
This theorem is referenced by:  splval  11482  splcl  11483  ida2  13907  coa2  13917  mamufval  27546  mapdhval  32536  hdmap1val  32611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-ot 3663  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-2nd 6139
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