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Theorem idval 25725
Description: Value of the identity function expressed with the  1st and  2nd functions. (Contributed by FL, 26-Oct-2007.)
Hypothesis
Ref Expression
idval.1  |-  J  =  ( id_ `  T
)
Assertion
Ref Expression
idval  |-  J  =  ( 1st `  ( 2nd `  T ) )

Proof of Theorem idval
StepHypRef Expression
1 idval.1 . 2  |-  J  =  ( id_ `  T
)
2 df-id_ 25719 . . . . 5  |-  id_  =  ( 1st  o.  2nd )
32fveq1i 5526 . . . 4  |-  ( id_ `  T )  =  ( ( 1st  o.  2nd ) `  T )
4 fo2nd 6140 . . . . . 6  |-  2nd : _V -onto-> _V
5 fof 5451 . . . . . 6  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
64, 5ax-mp 8 . . . . 5  |-  2nd : _V
--> _V
7 fvco3 5596 . . . . 5  |-  ( ( 2nd : _V --> _V  /\  T  e.  _V )  ->  ( ( 1st  o.  2nd ) `  T )  =  ( 1st `  ( 2nd `  T ) ) )
86, 7mpan 651 . . . 4  |-  ( T  e.  _V  ->  (
( 1st  o.  2nd ) `  T )  =  ( 1st `  ( 2nd `  T ) ) )
93, 8syl5eq 2327 . . 3  |-  ( T  e.  _V  ->  ( id_ `  T )  =  ( 1st `  ( 2nd `  T ) ) )
10 fvprc 5519 . . . 4  |-  ( -.  T  e.  _V  ->  ( id_ `  T )  =  (/) )
11 fvprc 5519 . . . . . 6  |-  ( -.  T  e.  _V  ->  ( 2nd `  T )  =  (/) )
1211fveq2d 5529 . . . . 5  |-  ( -.  T  e.  _V  ->  ( 1st `  ( 2nd `  T ) )  =  ( 1st `  (/) ) )
13 1st0 6126 . . . . 5  |-  ( 1st `  (/) )  =  (/)
1412, 13syl6req 2332 . . . 4  |-  ( -.  T  e.  _V  ->  (/)  =  ( 1st `  ( 2nd `  T ) ) )
1510, 14eqtrd 2315 . . 3  |-  ( -.  T  e.  _V  ->  ( id_ `  T )  =  ( 1st `  ( 2nd `  T ) ) )
169, 15pm2.61i 156 . 2  |-  ( id_ `  T )  =  ( 1st `  ( 2nd `  T ) )
171, 16eqtri 2303 1  |-  J  =  ( 1st `  ( 2nd `  T ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455    o. ccom 4693   -->wf 5251   -onto->wfo 5253   ` cfv 5255   1stc1st 6120   2ndc2nd 6121   id_cid_ 25714
This theorem is referenced by:  algi  25727  dedi  25737  dedalg  25743  cati  25755  catded  25764  issubcata  25846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-id_ 25719
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