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Theorem idfu2nd 14787
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 6094 . 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 14786 . . . . 5  |-  ( ph  ->  I  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  ( H `  z )
) ) >. )
76fveq2d 5695 . . . 4  |-  ( ph  ->  ( 2nd `  I
)  =  ( 2nd `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) ) >. )
)
8 fvex 5701 . . . . . . 7  |-  ( Base `  C )  e.  _V
93, 8eqeltri 2513 . . . . . 6  |-  B  e. 
_V
10 resiexg 6514 . . . . . 6  |-  ( B  e.  _V  ->  (  _I  |`  B )  e. 
_V )
119, 10ax-mp 5 . . . . 5  |-  (  _I  |`  B )  e.  _V
129, 9xpex 6508 . . . . . 6  |-  ( B  X.  B )  e. 
_V
1312mptex 5948 . . . . 5  |-  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `  z
) ) )  e. 
_V
1411, 13op2nd 6586 . . . 4  |-  ( 2nd `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) ) >. )  =  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) )
157, 14syl6eq 2491 . . 3  |-  ( ph  ->  ( 2nd `  I
)  =  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `  z
) ) ) )
16 simpr 461 . . . . . 6  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  z  =  <. X ,  Y >. )
1716fveq2d 5695 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( H `  <. X ,  Y >. ) )
18 df-ov 6094 . . . . 5  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
1917, 18syl6eqr 2493 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( X H Y ) )
2019reseq2d 5110 . . 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 4871 . . . 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 6116 . . . 4  |-  ( X H Y )  e. 
_V
26 resiexg 6514 . . . 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 5779 . 2  |-  ( ph  ->  ( ( 2nd `  I
) `  <. X ,  Y >. )  =  (  _I  |`  ( X H Y ) ) )
291, 28syl5eq 2487 1  |-  ( ph  ->  ( X ( 2nd `  I ) Y )  =  (  _I  |`  ( X H Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2972   <.cop 3883    e. cmpt 4350    _I cid 4631    X. cxp 4838    |` cres 4842   ` cfv 5418  (class class class)co 6091   2ndc2nd 6576   Basecbs 14174   Hom chom 14249   Catccat 14602  idfunccidfu 14765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-2nd 6578  df-idfu 14769
This theorem is referenced by:  idfu2  14788  idfucl  14791  cofulid  14800  cofurid  14801  idffth  14843  ressffth  14848  catciso  14975
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