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Theorem fullresc 14003
Description: The category formed by structure restriction is the same as the category restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
Hypotheses
Ref Expression
fullsubc.b  |-  B  =  ( Base `  C
)
fullsubc.h  |-  H  =  (  Homf 
`  C )
fullsubc.c  |-  ( ph  ->  C  e.  Cat )
fullsubc.s  |-  ( ph  ->  S  C_  B )
fullsubc.d  |-  D  =  ( Cs  S )
fullsubc.e  |-  E  =  ( C  |`cat  ( H  |`  ( S  X.  S
) ) )
Assertion
Ref Expression
fullresc  |-  ( ph  ->  ( (  Homf  `  D )  =  (  Homf  `  E )  /\  (compf `  D )  =  (compf `  E ) ) )

Proof of Theorem fullresc
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fullsubc.h . . . . . 6  |-  H  =  (  Homf 
`  C )
2 fullsubc.b . . . . . 6  |-  B  =  ( Base `  C
)
3 eqid 2404 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 fullsubc.s . . . . . . . 8  |-  ( ph  ->  S  C_  B )
54adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  S  C_  B )
6 simprl 733 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  S )
75, 6sseldd 3309 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  B )
8 simprr 734 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  S )
95, 8sseldd 3309 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  B )
101, 2, 3, 7, 9homfval 13873 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x H y )  =  ( x (  Hom  `  C
) y ) )
116, 8ovresd 6173 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( H  |`  ( S  X.  S
) ) y )  =  ( x H y ) )
12 fullsubc.e . . . . . . . 8  |-  E  =  ( C  |`cat  ( H  |`  ( S  X.  S
) ) )
13 fullsubc.c . . . . . . . 8  |-  ( ph  ->  C  e.  Cat )
141, 2homffn 13874 . . . . . . . . 9  |-  H  Fn  ( B  X.  B
)
15 xpss12 4940 . . . . . . . . . 10  |-  ( ( S  C_  B  /\  S  C_  B )  -> 
( S  X.  S
)  C_  ( B  X.  B ) )
164, 4, 15syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( S  X.  S
)  C_  ( B  X.  B ) )
17 fnssres 5517 . . . . . . . . 9  |-  ( ( H  Fn  ( B  X.  B )  /\  ( S  X.  S
)  C_  ( B  X.  B ) )  -> 
( H  |`  ( S  X.  S ) )  Fn  ( S  X.  S ) )
1814, 16, 17sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( H  |`  ( S  X.  S ) )  Fn  ( S  X.  S ) )
1912, 2, 13, 18, 4reschom 13985 . . . . . . 7  |-  ( ph  ->  ( H  |`  ( S  X.  S ) )  =  (  Hom  `  E
) )
2019proplem3 13871 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( H  |`  ( S  X.  S
) ) y )  =  ( x (  Hom  `  E )
y ) )
2111, 20eqtr3d 2438 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x H y )  =  ( x (  Hom  `  E
) y ) )
22 fullsubc.d . . . . . . . . . 10  |-  D  =  ( Cs  S )
2322, 2ressbas2 13475 . . . . . . . . 9  |-  ( S 
C_  B  ->  S  =  ( Base `  D
) )
244, 23syl 16 . . . . . . . 8  |-  ( ph  ->  S  =  ( Base `  D ) )
25 fvex 5701 . . . . . . . 8  |-  ( Base `  D )  e.  _V
2624, 25syl6eqel 2492 . . . . . . 7  |-  ( ph  ->  S  e.  _V )
2722, 3resshom 13601 . . . . . . 7  |-  ( S  e.  _V  ->  (  Hom  `  C )  =  (  Hom  `  D
) )
2826, 27syl 16 . . . . . 6  |-  ( ph  ->  (  Hom  `  C
)  =  (  Hom  `  D ) )
2928proplem3 13871 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x (  Hom  `  C ) y )  =  ( x (  Hom  `  D )
y ) )
3010, 21, 293eqtr3rd 2445 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x (  Hom  `  D ) y )  =  ( x (  Hom  `  E )
y ) )
3130ralrimivva 2758 . . 3  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x (  Hom  `  D ) y )  =  ( x (  Hom  `  E )
y ) )
32 eqid 2404 . . . 4  |-  (  Hom  `  D )  =  (  Hom  `  D )
33 eqid 2404 . . . 4  |-  (  Hom  `  E )  =  (  Hom  `  E )
3412, 2, 13, 18, 4rescbas 13984 . . . 4  |-  ( ph  ->  S  =  ( Base `  E ) )
3532, 33, 24, 34homfeq 13875 . . 3  |-  ( ph  ->  ( (  Homf  `  D )  =  (  Homf  `  E )  <->  A. x  e.  S  A. y  e.  S  ( x (  Hom  `  D ) y )  =  ( x (  Hom  `  E )
y ) ) )
3631, 35mpbird 224 . 2  |-  ( ph  ->  (  Homf 
`  D )  =  (  Homf 
`  E ) )
37 eqid 2404 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
3822, 37ressco 13602 . . . . 5  |-  ( S  e.  _V  ->  (comp `  C )  =  (comp `  D ) )
3926, 38syl 16 . . . 4  |-  ( ph  ->  (comp `  C )  =  (comp `  D )
)
4012, 2, 13, 18, 4, 37rescco 13987 . . . 4  |-  ( ph  ->  (comp `  C )  =  (comp `  E )
)
4139, 40eqtr3d 2438 . . 3  |-  ( ph  ->  (comp `  D )  =  (comp `  E )
)
4241, 36comfeqd 13888 . 2  |-  ( ph  ->  (compf `  D )  =  (compf `  E ) )
4336, 42jca 519 1  |-  ( ph  ->  ( (  Homf  `  D )  =  (  Homf  `  E )  /\  (compf `  D )  =  (compf `  E ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    C_ wss 3280    X. cxp 4835    |` cres 4839    Fn wfn 5408   ` cfv 5413  (class class class)co 6040   Basecbs 13424   ↾s cress 13425    Hom chom 13495  compcco 13496   Catccat 13844    Homf chomf 13846  compfccomf 13847    |`cat cresc 13963
This theorem is referenced by:  resscat  14004  funcres2c  14053  ressffth  14090  funcsetcres2  14203
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-hom 13508  df-cco 13509  df-homf 13850  df-comf 13851  df-resc 13966
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