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Theorem fullresc 15067
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 2460 . . . . . 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 3498 . . . . . 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 3498 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  B )
101, 2, 3, 7, 9homfval 14937 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x H y )  =  ( x ( Hom  `  C
) y ) )
116, 8ovresd 6418 . . . . . 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 14938 . . . . . . . . 9  |-  H  Fn  ( B  X.  B
)
15 xpss12 5099 . . . . . . . . . 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 5685 . . . . . . . . 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 15049 . . . . . . 7  |-  ( ph  ->  ( H  |`  ( S  X.  S ) )  =  ( Hom  `  E
) )
2019proplem3 14935 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( H  |`  ( S  X.  S
) ) y )  =  ( x ( Hom  `  E )
y ) )
2111, 20eqtr3d 2503 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x H y )  =  ( x ( Hom  `  E
) y ) )
22 fullsubc.d . . . . . . . . . 10  |-  D  =  ( Cs  S )
2322, 2ressbas2 14535 . . . . . . . . 9  |-  ( S 
C_  B  ->  S  =  ( Base `  D
) )
244, 23syl 16 . . . . . . . 8  |-  ( ph  ->  S  =  ( Base `  D ) )
25 fvex 5867 . . . . . . . 8  |-  ( Base `  D )  e.  _V
2624, 25syl6eqel 2556 . . . . . . 7  |-  ( ph  ->  S  e.  _V )
2722, 3resshom 14663 . . . . . . 7  |-  ( S  e.  _V  ->  ( Hom  `  C )  =  ( Hom  `  D
) )
2826, 27syl 16 . . . . . 6  |-  ( ph  ->  ( Hom  `  C
)  =  ( Hom  `  D ) )
2928proplem3 14935 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( Hom  `  C ) y )  =  ( x ( Hom  `  D )
y ) )
3010, 21, 293eqtr3rd 2510 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( Hom  `  D ) y )  =  ( x ( Hom  `  E )
y ) )
3130ralrimivva 2878 . . 3  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x ( Hom  `  D ) y )  =  ( x ( Hom  `  E )
y ) )
32 eqid 2460 . . . 4  |-  ( Hom  `  D )  =  ( Hom  `  D )
33 eqid 2460 . . . 4  |-  ( Hom  `  E )  =  ( Hom  `  E )
3412, 2, 13, 18, 4rescbas 15048 . . . 4  |-  ( ph  ->  S  =  ( Base `  E ) )
3532, 33, 24, 34homfeq 14939 . . 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 2460 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
3822, 37ressco 14664 . . . . 5  |-  ( S  e.  _V  ->  (comp `  C )  =  (comp `  D ) )
3926, 38syl 16 . . . 4  |-  ( ph  ->  (comp `  C )  =  (comp `  D )
)
4012, 2, 13, 18, 4, 37rescco 15051 . . . 4  |-  ( ph  ->  (comp `  C )  =  (comp `  E )
)
4139, 40eqtr3d 2503 . . 3  |-  ( ph  ->  (comp `  D )  =  (comp `  E )
)
4241, 36comfeqd 14952 . 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 1374    e. wcel 1762   A.wral 2807   _Vcvv 3106    C_ wss 3469    X. cxp 4990    |` cres 4994    Fn wfn 5574   ` cfv 5579  (class class class)co 6275   Basecbs 14479   ↾s cress 14480   Hom chom 14555  compcco 14556   Catccat 14908   Hom f chomf 14910  compfccomf 14911    |`cat cresc 15027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-hom 14568  df-cco 14569  df-homf 14914  df-comf 14915  df-resc 15030
This theorem is referenced by:  resscat  15068  funcres2c  15117  ressffth  15154  funcsetcres2  15267
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