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Theorem funcsetcres2 15281
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 2468 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( Hom f  `  E
)  =  ( Hom f  `  E ) )
2 eqidd 2468 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (compf `  E )  =  (compf `  E ) )
3 eqid 2467 . . . . . . . . 9  |-  ( Base `  C )  =  (
Base `  C )
4 eqid 2467 . . . . . . . . 9  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
5 resssetc.1 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  W )
6 resssetc.c . . . . . . . . . . . 12  |-  C  =  ( SetCat `  U )
76setccat 15273 . . . . . . . . . . 11  |-  ( U  e.  W  ->  C  e.  Cat )
85, 7syl 16 . . . . . . . . . 10  |-  ( ph  ->  C  e.  Cat )
98adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  C  e.  Cat )
10 resssetc.2 . . . . . . . . . . 11  |-  ( ph  ->  V  C_  U )
116, 5setcbas 15266 . . . . . . . . . . 11  |-  ( ph  ->  U  =  ( Base `  C ) )
1210, 11sseqtrd 3540 . . . . . . . . . 10  |-  ( ph  ->  V  C_  ( Base `  C ) )
1312adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  V  C_  ( Base `  C ) )
14 eqid 2467 . . . . . . . . 9  |-  ( Cs  V )  =  ( Cs  V )
15 eqid 2467 . . . . . . . . 9  |-  ( C  |`cat 
( ( Hom f  `  C )  |`  ( V  X.  V
) ) )  =  ( C  |`cat  ( ( Hom f  `  C )  |`  ( V  X.  V ) ) )
163, 4, 9, 13, 14, 15fullresc 15081 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( ( Hom f  `  ( Cs  V ) )  =  ( Hom f  `  ( C  |`cat  ( ( Hom f  `  C )  |`  ( V  X.  V
) ) ) )  /\  (compf `  ( Cs  V ) )  =  (compf `  ( C  |`cat  ( ( Hom f  `  C )  |`  ( V  X.  V ) ) ) ) ) )
1716simpld 459 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( Hom f  `  ( Cs  V ) )  =  ( Hom f  `  ( C  |`cat  ( ( Hom f  `  C )  |`  ( V  X.  V
) ) ) ) )
18 resssetc.d . . . . . . . . . 10  |-  D  =  ( SetCat `  V )
196, 18, 5, 10resssetc 15280 . . . . . . . . 9  |-  ( ph  ->  ( ( Hom f  `  ( Cs  V ) )  =  ( Hom f  `  D )  /\  (compf `  ( Cs  V ) )  =  (compf `  D ) ) )
2019adantr 465 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( ( Hom f  `  ( Cs  V ) )  =  ( Hom f  `  D )  /\  (compf `  ( Cs  V ) )  =  (compf `  D ) ) )
2120simpld 459 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( Hom f  `  ( Cs  V ) )  =  ( Hom f  `  D ) )
2217, 21eqtr3d 2510 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( Hom f  `  ( C  |`cat  ( ( Hom f  `  C )  |`  ( V  X.  V
) ) ) )  =  ( Hom f  `  D ) )
2316simprd 463 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (compf `  ( Cs  V ) )  =  (compf `  ( C  |`cat  ( ( Hom f  `  C )  |`  ( V  X.  V ) ) ) ) )
2420simprd 463 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (compf `  ( Cs  V ) )  =  (compf `  D ) )
2523, 24eqtr3d 2510 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (compf `  ( C  |`cat  ( ( Hom f  `  C )  |`  ( V  X.  V ) ) ) )  =  (compf `  D ) )
26 funcrcl 15093 . . . . . . . 8  |-  ( f  e.  ( E  Func  D )  ->  ( E  e.  Cat  /\  D  e. 
Cat ) )
2726adantl 466 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( E  e.  Cat  /\  D  e. 
Cat ) )
2827simpld 459 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  E  e.  Cat )
293, 4, 9, 13fullsubc 15080 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( ( Hom f  `  C )  |`  ( V  X.  V ) )  e.  (Subcat `  C
) )
3015, 29subccat 15078 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( C  |`cat  ( ( Hom f  `  C )  |`  ( V  X.  V
) ) )  e. 
Cat )
3127simprd 463 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  D  e.  Cat )
321, 2, 22, 25, 28, 28, 30, 31funcpropd 15130 . . . . 5  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( E  Func  ( C  |`cat  ( ( Hom f  `  C )  |`  ( V  X.  V ) ) ) )  =  ( E  Func  D )
)
33 funcres2 15128 . . . . . 6  |-  ( ( ( Hom f  `  C )  |`  ( V  X.  V
) )  e.  (Subcat `  C )  ->  ( E  Func  ( C  |`cat  ( ( Hom f  `  C )  |`  ( V  X.  V
) ) ) ) 
C_  ( E  Func  C ) )
3429, 33syl 16 . . . . 5  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( E  Func  ( C  |`cat  ( ( Hom f  `  C )  |`  ( V  X.  V ) ) ) )  C_  ( E  Func  C ) )
3532, 34eqsstr3d 3539 . . . 4  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( E  Func  D )  C_  ( E  Func  C ) )
36 simpr 461 . . . 4  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  f  e.  ( E  Func  D ) )
3735, 36sseldd 3505 . . 3  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  f  e.  ( E  Func  C ) )
3837ex 434 . 2  |-  ( ph  ->  ( f  e.  ( E  Func  D )  ->  f  e.  ( E 
Func  C ) ) )
3938ssrdv 3510 1  |-  ( ph  ->  ( E  Func  D
)  C_  ( E  Func  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476    X. cxp 4997    |` cres 5001   ` cfv 5588  (class class class)co 6285   Basecbs 14493   ↾s cress 14494   Catccat 14922   Hom f chomf 14924  compfccomf 14925    |`cat cresc 15041  Subcatcsubc 15042    Func cfunc 15084   SetCatcsetc 15263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  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 10978  df-uz 11084  df-fz 11674  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-hom 14582  df-cco 14583  df-cat 14926  df-cid 14927  df-homf 14928  df-comf 14929  df-ssc 15043  df-resc 15044  df-subc 15045  df-func 15088  df-setc 15264
This theorem is referenced by:  yonedalem1  15402
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