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Theorem fullresc 15257
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 2382 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
4 fullsubc.s . . . . . . . 8  |-  ( ph  ->  S  C_  B )
54adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  S  C_  B )
6 simprl 754 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  S )
75, 6sseldd 3418 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  B )
8 simprr 755 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  S )
95, 8sseldd 3418 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  B )
101, 2, 3, 7, 9homfval 15098 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x H y )  =  ( x ( Hom  `  C
) y ) )
116, 8ovresd 6342 . . . . . 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 15099 . . . . . . . . 9  |-  H  Fn  ( B  X.  B
)
15 xpss12 5021 . . . . . . . . . 10  |-  ( ( S  C_  B  /\  S  C_  B )  -> 
( S  X.  S
)  C_  ( B  X.  B ) )
164, 4, 15syl2anc 659 . . . . . . . . 9  |-  ( ph  ->  ( S  X.  S
)  C_  ( B  X.  B ) )
17 fnssres 5602 . . . . . . . . 9  |-  ( ( H  Fn  ( B  X.  B )  /\  ( S  X.  S
)  C_  ( B  X.  B ) )  -> 
( H  |`  ( S  X.  S ) )  Fn  ( S  X.  S ) )
1814, 16, 17sylancr 661 . . . . . . . 8  |-  ( ph  ->  ( H  |`  ( S  X.  S ) )  Fn  ( S  X.  S ) )
1912, 2, 13, 18, 4reschom 15236 . . . . . . 7  |-  ( ph  ->  ( H  |`  ( S  X.  S ) )  =  ( Hom  `  E
) )
2019oveqdr 6220 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( H  |`  ( S  X.  S
) ) y )  =  ( x ( Hom  `  E )
y ) )
2111, 20eqtr3d 2425 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x H y )  =  ( x ( Hom  `  E
) y ) )
22 fullsubc.d . . . . . . . . . 10  |-  D  =  ( Cs  S )
2322, 2ressbas2 14692 . . . . . . . . 9  |-  ( S 
C_  B  ->  S  =  ( Base `  D
) )
244, 23syl 16 . . . . . . . 8  |-  ( ph  ->  S  =  ( Base `  D ) )
25 fvex 5784 . . . . . . . 8  |-  ( Base `  D )  e.  _V
2624, 25syl6eqel 2478 . . . . . . 7  |-  ( ph  ->  S  e.  _V )
2722, 3resshom 14825 . . . . . . 7  |-  ( S  e.  _V  ->  ( Hom  `  C )  =  ( Hom  `  D
) )
2826, 27syl 16 . . . . . 6  |-  ( ph  ->  ( Hom  `  C
)  =  ( Hom  `  D ) )
2928oveqdr 6220 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( Hom  `  C ) y )  =  ( x ( Hom  `  D )
y ) )
3010, 21, 293eqtr3rd 2432 . . . 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 2382 . . . 4  |-  ( Hom  `  D )  =  ( Hom  `  D )
33 eqid 2382 . . . 4  |-  ( Hom  `  E )  =  ( Hom  `  E )
3412, 2, 13, 18, 4rescbas 15235 . . . 4  |-  ( ph  ->  S  =  ( Base `  E ) )
3532, 33, 24, 34homfeq 15100 . . 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 2382 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
3822, 37ressco 14826 . . . . 5  |-  ( S  e.  _V  ->  (comp `  C )  =  (comp `  D ) )
3926, 38syl 16 . . . 4  |-  ( ph  ->  (comp `  C )  =  (comp `  D )
)
4012, 2, 13, 18, 4, 37rescco 15238 . . . 4  |-  ( ph  ->  (comp `  C )  =  (comp `  E )
)
4139, 40eqtr3d 2425 . . 3  |-  ( ph  ->  (comp `  D )  =  (comp `  E )
)
4241, 36comfeqd 15113 . 2  |-  ( ph  ->  (compf `  D )  =  (compf `  E ) )
4336, 42jca 530 1  |-  ( ph  ->  ( ( Hom f  `  D )  =  ( Hom f  `  E )  /\  (compf `  D )  =  (compf `  E ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   A.wral 2732   _Vcvv 3034    C_ wss 3389    X. cxp 4911    |` cres 4915    Fn wfn 5491   ` cfv 5496  (class class class)co 6196   Basecbs 14634   ↾s cress 14635   Hom chom 14713  compcco 14714   Catccat 15071   Hom f chomf 15073  compfccomf 15074    |`cat cresc 15214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-hom 14726  df-cco 14727  df-homf 15077  df-comf 15078  df-resc 15217
This theorem is referenced by:  resscat  15258  funcres2c  15307  ressffth  15344  funcsetcres2  15489
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