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Theorem fullresc 13741
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 2296 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 fullsubc.s . . . . . . . 8  |-  ( ph  ->  S  C_  B )
54adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  S  C_  B )
6 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  S )
75, 6sseldd 3194 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  x  e.  B )
8 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  S )
95, 8sseldd 3194 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
y  e.  B )
101, 2, 3, 7, 9homfval 13611 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x H y )  =  ( x (  Hom  `  C
) y ) )
116, 8ovresd 6004 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( H  |`  ( S  X.  S
) ) y )  =  ( x H y ) )
12 fullsubc.e . . . . . . . . 9  |-  E  =  ( C  |`cat  ( H  |`  ( S  X.  S
) ) )
13 fullsubc.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
141, 2homffn 13612 . . . . . . . . . 10  |-  H  Fn  ( B  X.  B
)
15 xpss12 4808 . . . . . . . . . . 11  |-  ( ( S  C_  B  /\  S  C_  B )  -> 
( S  X.  S
)  C_  ( B  X.  B ) )
164, 4, 15syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( S  X.  S
)  C_  ( B  X.  B ) )
17 fnssres 5373 . . . . . . . . . 10  |-  ( ( H  Fn  ( B  X.  B )  /\  ( S  X.  S
)  C_  ( B  X.  B ) )  -> 
( H  |`  ( S  X.  S ) )  Fn  ( S  X.  S ) )
1814, 16, 17sylancr 644 . . . . . . . . 9  |-  ( ph  ->  ( H  |`  ( S  X.  S ) )  Fn  ( S  X.  S ) )
1912, 2, 13, 18, 4reschom 13723 . . . . . . . 8  |-  ( ph  ->  ( H  |`  ( S  X.  S ) )  =  (  Hom  `  E
) )
2019adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( H  |`  ( S  X.  S ) )  =  (  Hom  `  E
) )
2120oveqd 5891 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( H  |`  ( S  X.  S
) ) y )  =  ( x (  Hom  `  E )
y ) )
2211, 21eqtr3d 2330 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x H y )  =  ( x (  Hom  `  E
) y ) )
23 fullsubc.d . . . . . . . . . . 11  |-  D  =  ( Cs  S )
2423, 2ressbas2 13215 . . . . . . . . . 10  |-  ( S 
C_  B  ->  S  =  ( Base `  D
) )
254, 24syl 15 . . . . . . . . 9  |-  ( ph  ->  S  =  ( Base `  D ) )
26 fvex 5555 . . . . . . . . 9  |-  ( Base `  D )  e.  _V
2725, 26syl6eqel 2384 . . . . . . . 8  |-  ( ph  ->  S  e.  _V )
2823, 3resshom 13339 . . . . . . . 8  |-  ( S  e.  _V  ->  (  Hom  `  C )  =  (  Hom  `  D
) )
2927, 28syl 15 . . . . . . 7  |-  ( ph  ->  (  Hom  `  C
)  =  (  Hom  `  D ) )
3029adantr 451 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
(  Hom  `  C )  =  (  Hom  `  D
) )
3130oveqd 5891 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x (  Hom  `  C ) y )  =  ( x (  Hom  `  D )
y ) )
3210, 22, 313eqtr3rd 2337 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x (  Hom  `  D ) y )  =  ( x (  Hom  `  E )
y ) )
3332ralrimivva 2648 . . 3  |-  ( ph  ->  A. x  e.  S  A. y  e.  S  ( x (  Hom  `  D ) y )  =  ( x (  Hom  `  E )
y ) )
34 eqid 2296 . . . 4  |-  (  Hom  `  D )  =  (  Hom  `  D )
35 eqid 2296 . . . 4  |-  (  Hom  `  E )  =  (  Hom  `  E )
3612, 2, 13, 18, 4rescbas 13722 . . . 4  |-  ( ph  ->  S  =  ( Base `  E ) )
3734, 35, 25, 36homfeq 13613 . . 3  |-  ( ph  ->  ( (  Homf  `  D )  =  (  Homf  `  E )  <->  A. x  e.  S  A. y  e.  S  ( x (  Hom  `  D ) y )  =  ( x (  Hom  `  E )
y ) ) )
3833, 37mpbird 223 . 2  |-  ( ph  ->  (  Homf 
`  D )  =  (  Homf 
`  E ) )
39 eqid 2296 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
4023, 39ressco 13340 . . . . 5  |-  ( S  e.  _V  ->  (comp `  C )  =  (comp `  D ) )
4127, 40syl 15 . . . 4  |-  ( ph  ->  (comp `  C )  =  (comp `  D )
)
4212, 2, 13, 18, 4, 39rescco 13725 . . . 4  |-  ( ph  ->  (comp `  C )  =  (comp `  E )
)
4341, 42eqtr3d 2330 . . 3  |-  ( ph  ->  (comp `  D )  =  (comp `  E )
)
4443, 38comfeqd 13626 . 2  |-  ( ph  ->  (compf `  D )  =  (compf `  E ) )
4538, 44jca 518 1  |-  ( ph  ->  ( (  Homf  `  D )  =  (  Homf  `  E )  /\  (compf `  D )  =  (compf `  E ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165    X. cxp 4703    |` cres 4707    Fn wfn 5266   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165    Hom chom 13235  compcco 13236   Catccat 13582    Homf chomf 13584  compfccomf 13585    |`cat cresc 13701
This theorem is referenced by:  resscat  13742  funcres2c  13791  ressffth  13828  funcsetcres2  13941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-hom 13248  df-cco 13249  df-homf 13588  df-comf 13589  df-resc 13704
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