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Theorem fullres2c 15430
Description: Condition for a full functor to also be a full functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
Hypotheses
Ref Expression
ffthres2c.a  |-  A  =  ( Base `  C
)
ffthres2c.e  |-  E  =  ( Ds  S )
ffthres2c.d  |-  ( ph  ->  D  e.  Cat )
ffthres2c.r  |-  ( ph  ->  S  e.  V )
ffthres2c.1  |-  ( ph  ->  F : A --> S )
Assertion
Ref Expression
fullres2c  |-  ( ph  ->  ( F ( C Full 
D ) G  <->  F ( C Full  E ) G ) )

Proof of Theorem fullres2c
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffthres2c.a . . . 4  |-  A  =  ( Base `  C
)
2 ffthres2c.e . . . 4  |-  E  =  ( Ds  S )
3 ffthres2c.d . . . 4  |-  ( ph  ->  D  e.  Cat )
4 ffthres2c.r . . . 4  |-  ( ph  ->  S  e.  V )
5 ffthres2c.1 . . . 4  |-  ( ph  ->  F : A --> S )
61, 2, 3, 4, 5funcres2c 15392 . . 3  |-  ( ph  ->  ( F ( C 
Func  D ) G  <->  F ( C  Func  E ) G ) )
7 eqid 2454 . . . . . . . 8  |-  ( Hom  `  D )  =  ( Hom  `  D )
82, 7resshom 14910 . . . . . . 7  |-  ( S  e.  V  ->  ( Hom  `  D )  =  ( Hom  `  E
) )
94, 8syl 16 . . . . . 6  |-  ( ph  ->  ( Hom  `  D
)  =  ( Hom  `  E ) )
109oveqd 6287 . . . . 5  |-  ( ph  ->  ( ( F `  x ) ( Hom  `  D ) ( F `
 y ) )  =  ( ( F `
 x ) ( Hom  `  E )
( F `  y
) ) )
1110eqeq2d 2468 . . . 4  |-  ( ph  ->  ( ran  ( x G y )  =  ( ( F `  x ) ( Hom  `  D ) ( F `
 y ) )  <->  ran  ( x G y )  =  ( ( F `  x ) ( Hom  `  E
) ( F `  y ) ) ) )
12112ralbidv 2898 . . 3  |-  ( ph  ->  ( A. x  e.  A  A. y  e.  A  ran  ( x G y )  =  ( ( F `  x ) ( Hom  `  D ) ( F `
 y ) )  <->  A. x  e.  A  A. y  e.  A  ran  ( x G y )  =  ( ( F `  x ) ( Hom  `  E
) ( F `  y ) ) ) )
136, 12anbi12d 708 . 2  |-  ( ph  ->  ( ( F ( C  Func  D ) G  /\  A. x  e.  A  A. y  e.  A  ran  ( x G y )  =  ( ( F `  x ) ( Hom  `  D ) ( F `
 y ) ) )  <->  ( F ( C  Func  E ) G  /\  A. x  e.  A  A. y  e.  A  ran  ( x G y )  =  ( ( F `  x ) ( Hom  `  E ) ( F `
 y ) ) ) ) )
141, 7isfull 15401 . 2  |-  ( F ( C Full  D ) G  <->  ( F ( C  Func  D ) G  /\  A. x  e.  A  A. y  e.  A  ran  ( x G y )  =  ( ( F `  x ) ( Hom  `  D ) ( F `
 y ) ) ) )
15 eqid 2454 . . 3  |-  ( Hom  `  E )  =  ( Hom  `  E )
161, 15isfull 15401 . 2  |-  ( F ( C Full  E ) G  <->  ( F ( C  Func  E ) G  /\  A. x  e.  A  A. y  e.  A  ran  ( x G y )  =  ( ( F `  x ) ( Hom  `  E ) ( F `
 y ) ) ) )
1713, 14, 163bitr4g 288 1  |-  ( ph  ->  ( F ( C Full 
D ) G  <->  F ( C Full  E ) G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   class class class wbr 4439   ran crn 4989   -->wf 5566   ` cfv 5570  (class class class)co 6270   Basecbs 14719   ↾s cress 14720   Hom chom 14798   Catccat 15156    Func cfunc 15345   Full cful 15393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-hom 14811  df-cco 14812  df-cat 15160  df-cid 15161  df-homf 15162  df-comf 15163  df-ssc 15301  df-resc 15302  df-subc 15303  df-func 15349  df-full 15395
This theorem is referenced by:  ffthres2c  15431
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