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Theorem cmpval 25829
Description: Value of the identity function expressed with the  2nd functions. (Contributed by FL, 26-Oct-2007.)
Hypothesis
Ref Expression
cmpval.1  |-  G  =  ( o_ `  T
)
Assertion
Ref Expression
cmpval  |-  G  =  ( 2nd `  ( 2nd `  T ) )

Proof of Theorem cmpval
StepHypRef Expression
1 cmpval.1 . 2  |-  G  =  ( o_ `  T
)
2 df-cmpa 25823 . . . . 5  |-  o_  =  ( 2nd  o.  2nd )
32fveq1i 5542 . . . 4  |-  ( o_
`  T )  =  ( ( 2nd  o.  2nd ) `  T )
4 fo2nd 6156 . . . . . 6  |-  2nd : _V -onto-> _V
5 fof 5467 . . . . . 6  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
64, 5ax-mp 8 . . . . 5  |-  2nd : _V
--> _V
7 fvco3 5612 . . . . 5  |-  ( ( 2nd : _V --> _V  /\  T  e.  _V )  ->  ( ( 2nd  o.  2nd ) `  T )  =  ( 2nd `  ( 2nd `  T ) ) )
86, 7mpan 651 . . . 4  |-  ( T  e.  _V  ->  (
( 2nd  o.  2nd ) `  T )  =  ( 2nd `  ( 2nd `  T ) ) )
93, 8syl5eq 2340 . . 3  |-  ( T  e.  _V  ->  (
o_ `  T )  =  ( 2nd `  ( 2nd `  T ) ) )
10 fvprc 5535 . . . 4  |-  ( -.  T  e.  _V  ->  ( o_ `  T )  =  (/) )
11 fvprc 5535 . . . . . 6  |-  ( -.  T  e.  _V  ->  ( 2nd `  T )  =  (/) )
1211fveq2d 5545 . . . . 5  |-  ( -.  T  e.  _V  ->  ( 2nd `  ( 2nd `  T ) )  =  ( 2nd `  (/) ) )
13 2nd0 6143 . . . . 5  |-  ( 2nd `  (/) )  =  (/)
1412, 13syl6req 2345 . . . 4  |-  ( -.  T  e.  _V  ->  (/)  =  ( 2nd `  ( 2nd `  T ) ) )
1510, 14eqtrd 2328 . . 3  |-  ( -.  T  e.  _V  ->  ( o_ `  T )  =  ( 2nd `  ( 2nd `  T ) ) )
169, 15pm2.61i 156 . 2  |-  ( o_
`  T )  =  ( 2nd `  ( 2nd `  T ) )
171, 16eqtri 2316 1  |-  G  =  ( 2nd `  ( 2nd `  T ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468    o. ccom 4709   -->wf 5267   -onto->wfo 5269   ` cfv 5271   2ndc2nd 6137   o_co_ 25818
This theorem is referenced by:  algi  25830  dedi  25840  dedalg  25846  cati  25858  catded  25867  issubcata  25949
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-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-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-2nd 6139  df-cmpa 25823
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