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Theorem fullresc 15089
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 2441 . . . . . 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 3487 . . . . . 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 3487 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  B )
101, 2, 3, 7, 9homfval 14959 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x H y )  =  ( x ( Hom  `  C
) y ) )
116, 8ovresd 6424 . . . . . 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 14960 . . . . . . . . 9  |-  H  Fn  ( B  X.  B
)
15 xpss12 5094 . . . . . . . . . 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 5680 . . . . . . . . 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 15071 . . . . . . 7  |-  ( ph  ->  ( H  |`  ( S  X.  S ) )  =  ( Hom  `  E
) )
2019oveqdr 6301 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( H  |`  ( S  X.  S
) ) y )  =  ( x ( Hom  `  E )
y ) )
2111, 20eqtr3d 2484 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x H y )  =  ( x ( Hom  `  E
) y ) )
22 fullsubc.d . . . . . . . . . 10  |-  D  =  ( Cs  S )
2322, 2ressbas2 14560 . . . . . . . . 9  |-  ( S 
C_  B  ->  S  =  ( Base `  D
) )
244, 23syl 16 . . . . . . . 8  |-  ( ph  ->  S  =  ( Base `  D ) )
25 fvex 5862 . . . . . . . 8  |-  ( Base `  D )  e.  _V
2624, 25syl6eqel 2537 . . . . . . 7  |-  ( ph  ->  S  e.  _V )
2722, 3resshom 14688 . . . . . . 7  |-  ( S  e.  _V  ->  ( Hom  `  C )  =  ( Hom  `  D
) )
2826, 27syl 16 . . . . . 6  |-  ( ph  ->  ( Hom  `  C
)  =  ( Hom  `  D ) )
2928oveqdr 6301 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( Hom  `  C ) y )  =  ( x ( Hom  `  D )
y ) )
3010, 21, 293eqtr3rd 2491 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( Hom  `  D ) y )  =  ( x ( Hom  `  E )
y ) )
3130ralrimivva 2862 . . 3  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x ( Hom  `  D ) y )  =  ( x ( Hom  `  E )
y ) )
32 eqid 2441 . . . 4  |-  ( Hom  `  D )  =  ( Hom  `  D )
33 eqid 2441 . . . 4  |-  ( Hom  `  E )  =  ( Hom  `  E )
3412, 2, 13, 18, 4rescbas 15070 . . . 4  |-  ( ph  ->  S  =  ( Base `  E ) )
3532, 33, 24, 34homfeq 14961 . . 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 2441 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
3822, 37ressco 14689 . . . . 5  |-  ( S  e.  _V  ->  (comp `  C )  =  (comp `  D ) )
3926, 38syl 16 . . . 4  |-  ( ph  ->  (comp `  C )  =  (comp `  D )
)
4012, 2, 13, 18, 4, 37rescco 15073 . . . 4  |-  ( ph  ->  (comp `  C )  =  (comp `  E )
)
4139, 40eqtr3d 2484 . . 3  |-  ( ph  ->  (comp `  D )  =  (comp `  E )
)
4241, 36comfeqd 14974 . 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 1381    e. wcel 1802   A.wral 2791   _Vcvv 3093    C_ wss 3458    X. cxp 4983    |` cres 4987    Fn wfn 5569   ` cfv 5574  (class class class)co 6277   Basecbs 14504   ↾s cress 14505   Hom chom 14580  compcco 14581   Catccat 14933   Hom f chomf 14935  compfccomf 14936    |`cat cresc 15049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-hom 14593  df-cco 14594  df-homf 14939  df-comf 14940  df-resc 15052
This theorem is referenced by:  resscat  15090  funcres2c  15139  ressffth  15176  funcsetcres2  15289
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