MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fullresc Structured version   Unicode version

Theorem fullresc 14753
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  =  ( Hom f  `  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  ->  ( ( Hom f  `  D )  =  ( Hom f  `  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  =  ( Hom f  `  C )
2 fullsubc.b . . . . . 6  |-  B  =  ( Base `  C
)
3 eqid 2438 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
4 fullsubc.s . . . . . . . 8  |-  ( ph  ->  S  C_  B )
54adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  S  C_  B )
6 simprl 755 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  S )
75, 6sseldd 3352 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  B )
8 simprr 756 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  S )
95, 8sseldd 3352 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  B )
101, 2, 3, 7, 9homfval 14623 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x H y )  =  ( x ( Hom  `  C
) y ) )
116, 8ovresd 6226 . . . . . 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 14624 . . . . . . . . 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 661 . . . . . . . . 9  |-  ( ph  ->  ( S  X.  S
)  C_  ( B  X.  B ) )
17 fnssres 5519 . . . . . . . . 9  |-  ( ( H  Fn  ( B  X.  B )  /\  ( S  X.  S
)  C_  ( B  X.  B ) )  -> 
( H  |`  ( S  X.  S ) )  Fn  ( S  X.  S ) )
1814, 16, 17sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( H  |`  ( S  X.  S ) )  Fn  ( S  X.  S ) )
1912, 2, 13, 18, 4reschom 14735 . . . . . . 7  |-  ( ph  ->  ( H  |`  ( S  X.  S ) )  =  ( Hom  `  E
) )
2019proplem3 14621 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( H  |`  ( S  X.  S
) ) y )  =  ( x ( Hom  `  E )
y ) )
2111, 20eqtr3d 2472 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x H y )  =  ( x ( Hom  `  E
) y ) )
22 fullsubc.d . . . . . . . . . 10  |-  D  =  ( Cs  S )
2322, 2ressbas2 14221 . . . . . . . . 9  |-  ( S 
C_  B  ->  S  =  ( Base `  D
) )
244, 23syl 16 . . . . . . . 8  |-  ( ph  ->  S  =  ( Base `  D ) )
25 fvex 5696 . . . . . . . 8  |-  ( Base `  D )  e.  _V
2624, 25syl6eqel 2526 . . . . . . 7  |-  ( ph  ->  S  e.  _V )
2722, 3resshom 14349 . . . . . . 7  |-  ( S  e.  _V  ->  ( Hom  `  C )  =  ( Hom  `  D
) )
2826, 27syl 16 . . . . . 6  |-  ( ph  ->  ( Hom  `  C
)  =  ( Hom  `  D ) )
2928proplem3 14621 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( Hom  `  C ) y )  =  ( x ( Hom  `  D )
y ) )
3010, 21, 293eqtr3rd 2479 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( Hom  `  D ) y )  =  ( x ( Hom  `  E )
y ) )
3130ralrimivva 2803 . . 3  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x ( Hom  `  D ) y )  =  ( x ( Hom  `  E )
y ) )
32 eqid 2438 . . . 4  |-  ( Hom  `  D )  =  ( Hom  `  D )
33 eqid 2438 . . . 4  |-  ( Hom  `  E )  =  ( Hom  `  E )
3412, 2, 13, 18, 4rescbas 14734 . . . 4  |-  ( ph  ->  S  =  ( Base `  E ) )
3532, 33, 24, 34homfeq 14625 . . 3  |-  ( ph  ->  ( ( Hom f  `  D )  =  ( Hom f  `  E )  <->  A. x  e.  S  A. y  e.  S  ( x ( Hom  `  D ) y )  =  ( x ( Hom  `  E )
y ) ) )
3631, 35mpbird 232 . 2  |-  ( ph  ->  ( Hom f  `  D )  =  ( Hom f  `  E ) )
37 eqid 2438 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
3822, 37ressco 14350 . . . . 5  |-  ( S  e.  _V  ->  (comp `  C )  =  (comp `  D ) )
3926, 38syl 16 . . . 4  |-  ( ph  ->  (comp `  C )  =  (comp `  D )
)
4012, 2, 13, 18, 4, 37rescco 14737 . . . 4  |-  ( ph  ->  (comp `  C )  =  (comp `  E )
)
4139, 40eqtr3d 2472 . . 3  |-  ( ph  ->  (comp `  D )  =  (comp `  E )
)
4241, 36comfeqd 14638 . 2  |-  ( ph  ->  (compf `  D )  =  (compf `  E ) )
4336, 42jca 532 1  |-  ( ph  ->  ( ( Hom f  `  D )  =  ( Hom f  `  E )  /\  (compf `  D )  =  (compf `  E ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710   _Vcvv 2967    C_ wss 3323    X. cxp 4833    |` cres 4837    Fn wfn 5408   ` cfv 5413  (class class class)co 6086   Basecbs 14166   ↾s cress 14167   Hom chom 14241  compcco 14242   Catccat 14594   Hom f chomf 14596  compfccomf 14597    |`cat cresc 14713
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-hom 14254  df-cco 14255  df-homf 14600  df-comf 14601  df-resc 14716
This theorem is referenced by:  resscat  14754  funcres2c  14803  ressffth  14840  funcsetcres2  14953
  Copyright terms: Public domain W3C validator