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Theorem idfu2nd 13767
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 5877 . 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 13766 . . . . 5  |-  ( ph  ->  I  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  ( H `  z )
) ) >. )
76fveq2d 5545 . . . 4  |-  ( ph  ->  ( 2nd `  I
)  =  ( 2nd `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) ) >. )
)
8 fvex 5555 . . . . . . 7  |-  ( Base `  C )  e.  _V
93, 8eqeltri 2366 . . . . . 6  |-  B  e. 
_V
10 resiexg 5013 . . . . . 6  |-  ( B  e.  _V  ->  (  _I  |`  B )  e. 
_V )
119, 10ax-mp 8 . . . . 5  |-  (  _I  |`  B )  e.  _V
129, 9xpex 4817 . . . . . 6  |-  ( B  X.  B )  e. 
_V
1312mptex 5762 . . . . 5  |-  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `  z
) ) )  e. 
_V
1411, 13op2nd 6145 . . . 4  |-  ( 2nd `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) ) >. )  =  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) )
157, 14syl6eq 2344 . . 3  |-  ( ph  ->  ( 2nd `  I
)  =  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `  z
) ) ) )
16 simpr 447 . . . . . 6  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  z  =  <. X ,  Y >. )
1716fveq2d 5545 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( H `  <. X ,  Y >. ) )
18 df-ov 5877 . . . . 5  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
1917, 18syl6eqr 2346 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( H `  z )  =  ( X H Y ) )
2019reseq2d 4971 . . 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 4737 . . . 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 5899 . . . 4  |-  ( X H Y )  e. 
_V
26 resiexg 5013 . . . 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 5622 . 2  |-  ( ph  ->  ( ( 2nd `  I
) `  <. X ,  Y >. )  =  (  _I  |`  ( X H Y ) ) )
291, 28syl5eq 2340 1  |-  ( ph  ->  ( X ( 2nd `  I ) Y )  =  (  _I  |`  ( X H Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    e. cmpt 4093    _I cid 4320    X. cxp 4703    |` cres 4707   ` cfv 5271  (class class class)co 5874   2ndc2nd 6137   Basecbs 13164    Hom chom 13235   Catccat 13582  idfunccidfu 13745
This theorem is referenced by:  idfu2  13768  idfucl  13771  cofulid  13780  cofurid  13781  idffth  13823  ressffth  13828  catciso  13955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-2nd 6139  df-idfu 13749
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