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Theorem idfu2nd 13751
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 5861 . 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 13750 . . . . 5  |-  ( ph  ->  I  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  ( H `  z )
) ) >. )
76fveq2d 5529 . . . 4  |-  ( ph  ->  ( 2nd `  I
)  =  ( 2nd `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) ) >. )
)
8 fvex 5539 . . . . . . 7  |-  ( Base `  C )  e.  _V
93, 8eqeltri 2353 . . . . . 6  |-  B  e. 
_V
10 resiexg 4997 . . . . . 6  |-  ( B  e.  _V  ->  (  _I  |`  B )  e. 
_V )
119, 10ax-mp 8 . . . . 5  |-  (  _I  |`  B )  e.  _V
129, 9xpex 4801 . . . . . 6  |-  ( B  X.  B )  e. 
_V
1312mptex 5746 . . . . 5  |-  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `  z
) ) )  e. 
_V
1411, 13op2nd 6129 . . . 4  |-  ( 2nd `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) ) >. )  =  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) )
157, 14syl6eq 2331 . . 3  |-  ( ph  ->  ( 2nd `  I
)  =  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `  z
) ) ) )
16 simpr 447 . . . . . 6  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  z  =  <. X ,  Y >. )
1716fveq2d 5529 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( H `  <. X ,  Y >. ) )
18 df-ov 5861 . . . . 5  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
1917, 18syl6eqr 2333 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( X H Y ) )
2019reseq2d 4955 . . 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 4721 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
2421, 22, 23syl2anc 642 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
25 ovex 5883 . . . 4  |-  ( X H Y )  e. 
_V
26 resiexg 4997 . . . 4  |-  ( ( X H Y )  e.  _V  ->  (  _I  |`  ( X H Y ) )  e. 
_V )
2725, 26mp1i 11 . . 3  |-  ( ph  ->  (  _I  |`  ( X H Y ) )  e.  _V )
2815, 20, 24, 27fvmptd 5606 . 2  |-  ( ph  ->  ( ( 2nd `  I
) `  <. X ,  Y >. )  =  (  _I  |`  ( X H Y ) ) )
291, 28syl5eq 2327 1  |-  ( ph  ->  ( X ( 2nd `  I ) Y )  =  (  _I  |`  ( X H Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    e. cmpt 4077    _I cid 4304    X. cxp 4687    |` cres 4691   ` cfv 5255  (class class class)co 5858   2ndc2nd 6121   Basecbs 13148    Hom chom 13219   Catccat 13566  idfunccidfu 13729
This theorem is referenced by:  idfu2  13752  idfucl  13755  cofulid  13764  cofurid  13765  idffth  13807  ressffth  13812  catciso  13939
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-rep 4131  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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-2nd 6123  df-idfu 13733
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