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Theorem idfu2nd 14076
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 6086 . 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 14075 . . . . 5  |-  ( ph  ->  I  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  ( H `  z )
) ) >. )
76fveq2d 5734 . . . 4  |-  ( ph  ->  ( 2nd `  I
)  =  ( 2nd `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) ) >. )
)
8 fvex 5744 . . . . . . 7  |-  ( Base `  C )  e.  _V
93, 8eqeltri 2508 . . . . . 6  |-  B  e. 
_V
10 resiexg 5190 . . . . . 6  |-  ( B  e.  _V  ->  (  _I  |`  B )  e. 
_V )
119, 10ax-mp 8 . . . . 5  |-  (  _I  |`  B )  e.  _V
129, 9xpex 4992 . . . . . 6  |-  ( B  X.  B )  e. 
_V
1312mptex 5968 . . . . 5  |-  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `  z
) ) )  e. 
_V
1411, 13op2nd 6358 . . . 4  |-  ( 2nd `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) ) >. )  =  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) )
157, 14syl6eq 2486 . . 3  |-  ( ph  ->  ( 2nd `  I
)  =  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `  z
) ) ) )
16 simpr 449 . . . . . 6  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  z  =  <. X ,  Y >. )
1716fveq2d 5734 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( H `  <. X ,  Y >. ) )
18 df-ov 6086 . . . . 5  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
1917, 18syl6eqr 2488 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( X H Y ) )
2019reseq2d 5148 . . 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 4912 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
2421, 22, 23syl2anc 644 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
25 ovex 6108 . . . 4  |-  ( X H Y )  e. 
_V
26 resiexg 5190 . . . 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 5812 . 2  |-  ( ph  ->  ( ( 2nd `  I
) `  <. X ,  Y >. )  =  (  _I  |`  ( X H Y ) ) )
291, 28syl5eq 2482 1  |-  ( ph  ->  ( X ( 2nd `  I ) Y )  =  (  _I  |`  ( X H Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819    e. cmpt 4268    _I cid 4495    X. cxp 4878    |` cres 4882   ` cfv 5456  (class class class)co 6083   2ndc2nd 6350   Basecbs 13471    Hom chom 13542   Catccat 13891  idfunccidfu 14054
This theorem is referenced by:  idfu2  14077  idfucl  14080  cofulid  14089  cofurid  14090  idffth  14132  ressffth  14137  catciso  14264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-2nd 6352  df-idfu 14058
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