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Theorem funcsetcres2 14240
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 2436 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (  Homf  `  E
)  =  (  Homf  `  E ) )
2 eqidd 2436 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (compf `  E )  =  (compf `  E ) )
3 eqid 2435 . . . . . . . . 9  |-  ( Base `  C )  =  (
Base `  C )
4 eqid 2435 . . . . . . . . 9  |-  (  Homf  `  C )  =  (  Homf 
`  C )
5 resssetc.1 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  W )
6 resssetc.c . . . . . . . . . . . 12  |-  C  =  ( SetCat `  U )
76setccat 14232 . . . . . . . . . . 11  |-  ( U  e.  W  ->  C  e.  Cat )
85, 7syl 16 . . . . . . . . . 10  |-  ( ph  ->  C  e.  Cat )
98adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  C  e.  Cat )
10 resssetc.2 . . . . . . . . . . 11  |-  ( ph  ->  V  C_  U )
116, 5setcbas 14225 . . . . . . . . . . 11  |-  ( ph  ->  U  =  ( Base `  C ) )
1210, 11sseqtrd 3376 . . . . . . . . . 10  |-  ( ph  ->  V  C_  ( Base `  C ) )
1312adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  V  C_  ( Base `  C ) )
14 eqid 2435 . . . . . . . . 9  |-  ( Cs  V )  =  ( Cs  V )
15 eqid 2435 . . . . . . . . 9  |-  ( C  |`cat 
( (  Homf  `  C )  |`  ( V  X.  V
) ) )  =  ( C  |`cat  ( (  Homf  `  C )  |`  ( V  X.  V ) ) )
163, 4, 9, 13, 14, 15fullresc 14040 . . . . . . . 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 ) ) ) ) ) )
1716simpld 446 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (  Homf  `  ( Cs  V ) )  =  (  Homf 
`  ( C  |`cat  ( (  Homf 
`  C )  |`  ( V  X.  V
) ) ) ) )
18 resssetc.d . . . . . . . . . 10  |-  D  =  ( SetCat `  V )
196, 18, 5, 10resssetc 14239 . . . . . . . . 9  |-  ( ph  ->  ( (  Homf  `  ( Cs  V ) )  =  (  Homf 
`  D )  /\  (compf `  ( Cs  V ) )  =  (compf `  D ) ) )
2019adantr 452 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( (  Homf  `  ( Cs  V ) )  =  (  Homf 
`  D )  /\  (compf `  ( Cs  V ) )  =  (compf `  D ) ) )
2120simpld 446 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (  Homf  `  ( Cs  V ) )  =  (  Homf 
`  D ) )
2217, 21eqtr3d 2469 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (  Homf  `  ( C  |`cat  ( (  Homf  `  C )  |`  ( V  X.  V
) ) ) )  =  (  Homf  `  D ) )
2316simprd 450 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (compf `  ( Cs  V ) )  =  (compf `  ( C  |`cat  ( (  Homf  `  C )  |`  ( V  X.  V ) ) ) ) )
2420simprd 450 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (compf `  ( Cs  V ) )  =  (compf `  D ) )
2523, 24eqtr3d 2469 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  (compf `  ( C  |`cat  ( (  Homf  `  C )  |`  ( V  X.  V ) ) ) )  =  (compf `  D ) )
26 funcrcl 14052 . . . . . . . 8  |-  ( f  e.  ( E  Func  D )  ->  ( E  e.  Cat  /\  D  e. 
Cat ) )
2726adantl 453 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( E  e.  Cat  /\  D  e. 
Cat ) )
2827simpld 446 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  E  e.  Cat )
293, 4, 9, 13fullsubc 14039 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( (  Homf  `  C )  |`  ( V  X.  V ) )  e.  (Subcat `  C
) )
3015, 29subccat 14037 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( C  |`cat  ( (  Homf 
`  C )  |`  ( V  X.  V
) ) )  e. 
Cat )
3127simprd 450 . . . . . 6  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  D  e.  Cat )
321, 2, 22, 25, 28, 28, 30, 31funcpropd 14089 . . . . 5  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( E  Func  ( C  |`cat  ( (  Homf  `  C )  |`  ( V  X.  V ) ) ) )  =  ( E  Func  D )
)
33 funcres2 14087 . . . . . 6  |-  ( ( (  Homf 
`  C )  |`  ( V  X.  V
) )  e.  (Subcat `  C )  ->  ( E  Func  ( C  |`cat  ( (  Homf 
`  C )  |`  ( V  X.  V
) ) ) ) 
C_  ( E  Func  C ) )
3429, 33syl 16 . . . . 5  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( E  Func  ( C  |`cat  ( (  Homf  `  C )  |`  ( V  X.  V ) ) ) )  C_  ( E  Func  C ) )
3532, 34eqsstr3d 3375 . . . 4  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  ( E  Func  D )  C_  ( E  Func  C ) )
36 simpr 448 . . . 4  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  f  e.  ( E  Func  D ) )
3735, 36sseldd 3341 . . 3  |-  ( (
ph  /\  f  e.  ( E  Func  D ) )  ->  f  e.  ( E  Func  C ) )
3837ex 424 . 2  |-  ( ph  ->  ( f  e.  ( E  Func  D )  ->  f  e.  ( E 
Func  C ) ) )
3938ssrdv 3346 1  |-  ( ph  ->  ( E  Func  D
)  C_  ( E  Func  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312    X. cxp 4868    |` cres 4872   ` cfv 5446  (class class class)co 6073   Basecbs 13461   ↾s cress 13462   Catccat 13881    Homf chomf 13883  compfccomf 13884    |`cat cresc 14000  Subcatcsubc 14001    Func cfunc 14043   SetCatcsetc 14222
This theorem is referenced by:  yonedalem1  14361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-hom 13545  df-cco 13546  df-cat 13885  df-cid 13886  df-homf 13887  df-comf 13888  df-ssc 14002  df-resc 14003  df-subc 14004  df-func 14047  df-setc 14223
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