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Theorem idfu2nd 15120
Description: Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
idfuval.i  |-  I  =  (idfunc `  C )
idfuval.b  |-  B  =  ( Base `  C
)
idfuval.c  |-  ( ph  ->  C  e.  Cat )
idfuval.h  |-  H  =  ( Hom  `  C
)
idfu2nd.x  |-  ( ph  ->  X  e.  B )
idfu2nd.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
idfu2nd  |-  ( ph  ->  ( X ( 2nd `  I ) Y )  =  (  _I  |`  ( X H Y ) ) )

Proof of Theorem idfu2nd
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-ov 6298 . 2  |-  ( X ( 2nd `  I
) Y )  =  ( ( 2nd `  I
) `  <. X ,  Y >. )
2 idfuval.i . . . . . 6  |-  I  =  (idfunc `  C )
3 idfuval.b . . . . . 6  |-  B  =  ( Base `  C
)
4 idfuval.c . . . . . 6  |-  ( ph  ->  C  e.  Cat )
5 idfuval.h . . . . . 6  |-  H  =  ( Hom  `  C
)
62, 3, 4, 5idfuval 15119 . . . . 5  |-  ( ph  ->  I  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  ( H `  z )
) ) >. )
76fveq2d 5876 . . . 4  |-  ( ph  ->  ( 2nd `  I
)  =  ( 2nd `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) ) >. )
)
8 fvex 5882 . . . . . . 7  |-  ( Base `  C )  e.  _V
93, 8eqeltri 2551 . . . . . 6  |-  B  e. 
_V
10 resiexg 6731 . . . . . 6  |-  ( B  e.  _V  ->  (  _I  |`  B )  e. 
_V )
119, 10ax-mp 5 . . . . 5  |-  (  _I  |`  B )  e.  _V
129, 9xpex 6599 . . . . . 6  |-  ( B  X.  B )  e. 
_V
1312mptex 6142 . . . . 5  |-  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `  z
) ) )  e. 
_V
1411, 13op2nd 6804 . . . 4  |-  ( 2nd `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) ) >. )  =  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) )
157, 14syl6eq 2524 . . 3  |-  ( ph  ->  ( 2nd `  I
)  =  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `  z
) ) ) )
16 simpr 461 . . . . . 6  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  z  =  <. X ,  Y >. )
1716fveq2d 5876 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( H `  <. X ,  Y >. ) )
18 df-ov 6298 . . . . 5  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
1917, 18syl6eqr 2526 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( X H Y ) )
2019reseq2d 5279 . . 3  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  (  _I  |`  ( H `  z )
)  =  (  _I  |`  ( X H Y ) ) )
21 idfu2nd.x . . . 4  |-  ( ph  ->  X  e.  B )
22 idfu2nd.y . . . 4  |-  ( ph  ->  Y  e.  B )
23 opelxpi 5037 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
2421, 22, 23syl2anc 661 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
25 ovex 6320 . . . 4  |-  ( X H Y )  e. 
_V
26 resiexg 6731 . . . 4  |-  ( ( X H Y )  e.  _V  ->  (  _I  |`  ( X H Y ) )  e. 
_V )
2725, 26mp1i 12 . . 3  |-  ( ph  ->  (  _I  |`  ( X H Y ) )  e.  _V )
2815, 20, 24, 27fvmptd 5962 . 2  |-  ( ph  ->  ( ( 2nd `  I
) `  <. X ,  Y >. )  =  (  _I  |`  ( X H Y ) ) )
291, 28syl5eq 2520 1  |-  ( ph  ->  ( X ( 2nd `  I ) Y )  =  (  _I  |`  ( X H Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   <.cop 4039    |-> cmpt 4511    _I cid 4796    X. cxp 5003    |` cres 5007   ` cfv 5594  (class class class)co 6295   2ndc2nd 6794   Basecbs 14506   Hom chom 14582   Catccat 14935  idfunccidfu 15098
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-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-2nd 6796  df-idfu 15102
This theorem is referenced by:  idfu2  15121  idfucl  15124  cofulid  15133  cofurid  15134  idffth  15176  ressffth  15181  catciso  15308
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