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Theorem funcsetcres2 13941
Description: A functor into a smaller category of sets is a functor into the larger category. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
resssetc.c  |-  C  =  ( SetCat `  U )
resssetc.d  |-  D  =  ( SetCat `  V )
resssetc.1  |-  ( ph  ->  U  e.  W )
resssetc.2  |-  ( ph  ->  V  C_  U )
Assertion
Ref Expression
funcsetcres2  |-  ( ph  ->  ( E  Func  D
)  C_  ( E  Func  C ) )

Proof of Theorem funcsetcres2
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 eqidd 2297 . . . . . . 7  |-  ( ph  ->  (  Homf 
`  E )  =  (  Homf 
`  E ) )
21adantr 451 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (  Homf  `  E
)  =  (  Homf  `  E ) )
3 eqidd 2297 . . . . . . 7  |-  ( ph  ->  (compf `  E )  =  (compf `  E ) )
43adantr 451 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (compf `  E )  =  (compf `  E ) )
5 eqid 2296 . . . . . . . . 9  |-  ( Base `  C )  =  (
Base `  C )
6 eqid 2296 . . . . . . . . 9  |-  (  Homf  `  C )  =  (  Homf 
`  C )
7 resssetc.1 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  W )
8 resssetc.c . . . . . . . . . . . 12  |-  C  =  ( SetCat `  U )
98setccat 13933 . . . . . . . . . . 11  |-  ( U  e.  W  ->  C  e.  Cat )
107, 9syl 15 . . . . . . . . . 10  |-  ( ph  ->  C  e.  Cat )
1110adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  C  e.  Cat )
12 resssetc.2 . . . . . . . . . . 11  |-  ( ph  ->  V  C_  U )
138, 7setcbas 13926 . . . . . . . . . . 11  |-  ( ph  ->  U  =  ( Base `  C ) )
1412, 13sseqtrd 3227 . . . . . . . . . 10  |-  ( ph  ->  V  C_  ( Base `  C ) )
1514adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  V  C_  ( Base `  C ) )
16 eqid 2296 . . . . . . . . 9  |-  ( Cs  V )  =  ( Cs  V )
17 eqid 2296 . . . . . . . . 9  |-  ( C  |`cat 
( (  Homf  `  C )  |`  ( V  X.  V
) ) )  =  ( C  |`cat  ( (  Homf  `  C )  |`  ( V  X.  V ) ) )
185, 6, 11, 15, 16, 17fullresc 13741 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( (  Homf  `  ( Cs  V ) )  =  (  Homf 
`  ( C  |`cat  ( (  Homf 
`  C )  |`  ( V  X.  V
) ) ) )  /\  (compf `  ( Cs  V ) )  =  (compf `  ( C  |`cat  ( (  Homf  `  C )  |`  ( V  X.  V ) ) ) ) ) )
1918simpld 445 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (  Homf  `  ( Cs  V ) )  =  (  Homf 
`  ( C  |`cat  ( (  Homf 
`  C )  |`  ( V  X.  V
) ) ) ) )
20 resssetc.d . . . . . . . . . 10  |-  D  =  ( SetCat `  V )
218, 20, 7, 12resssetc 13940 . . . . . . . . 9  |-  ( ph  ->  ( (  Homf  `  ( Cs  V ) )  =  (  Homf 
`  D )  /\  (compf `  ( Cs  V ) )  =  (compf `  D ) ) )
2221adantr 451 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( (  Homf  `  ( Cs  V ) )  =  (  Homf 
`  D )  /\  (compf `  ( Cs  V ) )  =  (compf `  D ) ) )
2322simpld 445 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (  Homf  `  ( Cs  V ) )  =  (  Homf 
`  D ) )
2419, 23eqtr3d 2330 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (  Homf  `  ( C  |`cat  ( (  Homf  `  C )  |`  ( V  X.  V
) ) ) )  =  (  Homf  `  D ) )
2518simprd 449 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (compf `  ( Cs  V ) )  =  (compf `  ( C  |`cat  ( (  Homf  `  C )  |`  ( V  X.  V ) ) ) ) )
2622simprd 449 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (compf `  ( Cs  V ) )  =  (compf `  D ) )
2725, 26eqtr3d 2330 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (compf `  ( C  |`cat  ( (  Homf  `  C )  |`  ( V  X.  V ) ) ) )  =  (compf `  D ) )
28 funcrcl 13753 . . . . . . . 8  |-  ( f  e.  ( E  Func  D )  ->  ( E  e.  Cat  /\  D  e. 
Cat ) )
2928adantl 452 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( E  e.  Cat  /\  D  e. 
Cat ) )
3029simpld 445 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  E  e.  Cat )
315, 6, 11, 15fullsubc 13740 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( (  Homf  `  C )  |`  ( V  X.  V ) )  e.  (Subcat `  C
) )
3217, 31subccat 13738 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( C  |`cat  ( (  Homf 
`  C )  |`  ( V  X.  V
) ) )  e. 
Cat )
3329simprd 449 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  D  e.  Cat )
342, 4, 24, 27, 30, 30, 32, 33funcpropd 13790 . . . . 5  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( E  Func  ( C  |`cat  ( (  Homf  `  C )  |`  ( V  X.  V ) ) ) )  =  ( E  Func  D )
)
35 funcres2 13788 . . . . . 6  |-  ( ( (  Homf 
`  C )  |`  ( V  X.  V
) )  e.  (Subcat `  C )  ->  ( E  Func  ( C  |`cat  ( (  Homf 
`  C )  |`  ( V  X.  V
) ) ) ) 
C_  ( E  Func  C ) )
3631, 35syl 15 . . . . 5  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( E  Func  ( C  |`cat  ( (  Homf  `  C )  |`  ( V  X.  V ) ) ) )  C_  ( E  Func  C ) )
3734, 36eqsstr3d 3226 . . . 4  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( E  Func  D )  C_  ( E  Func  C ) )
38 simpr 447 . . . 4  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  f  e.  ( E  Func  D ) )
3937, 38sseldd 3194 . . 3  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  f  e.  ( E  Func  C ) )
4039ex 423 . 2  |-  ( ph  ->  ( f  e.  ( E  Func  D )  ->  f  e.  ( E 
Func  C ) ) )
4140ssrdv 3198 1  |-  ( ph  ->  ( E  Func  D
)  C_  ( E  Func  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165    X. cxp 4703    |` cres 4707   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   Catccat 13582    Homf chomf 13584  compfccomf 13585    |`cat cresc 13701  Subcatcsubc 13702    Func cfunc 13744   SetCatcsetc 13923
This theorem is referenced by:  yonedalem1  14062
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-rmo 2564  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-int 3879  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-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  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-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-hom 13248  df-cco 13249  df-cat 13586  df-cid 13587  df-homf 13588  df-comf 13589  df-ssc 13703  df-resc 13704  df-subc 13705  df-func 13748  df-setc 13924
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