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Theorem reschomf 13958
Description: Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rescbas.d  |-  D  =  ( C  |`cat  H )
rescbas.b  |-  B  =  ( Base `  C
)
rescbas.c  |-  ( ph  ->  C  e.  V )
rescbas.h  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
rescbas.s  |-  ( ph  ->  S  C_  B )
Assertion
Ref Expression
reschomf  |-  ( ph  ->  H  =  (  Homf  `  D ) )

Proof of Theorem reschomf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rescbas.d . . . 4  |-  D  =  ( C  |`cat  H )
2 rescbas.b . . . 4  |-  B  =  ( Base `  C
)
3 rescbas.c . . . 4  |-  ( ph  ->  C  e.  V )
4 rescbas.h . . . 4  |-  ( ph  ->  H  Fn  ( S  X.  S ) )
5 rescbas.s . . . 4  |-  ( ph  ->  S  C_  B )
61, 2, 3, 4, 5reschom 13957 . . 3  |-  ( ph  ->  H  =  (  Hom  `  D ) )
71, 2, 3, 4, 5rescbas 13956 . . . . . . 7  |-  ( ph  ->  S  =  ( Base `  D ) )
87, 7xpeq12d 4843 . . . . . 6  |-  ( ph  ->  ( S  X.  S
)  =  ( (
Base `  D )  X.  ( Base `  D
) ) )
96, 8fneq12d 5478 . . . . 5  |-  ( ph  ->  ( H  Fn  ( S  X.  S )  <->  (  Hom  `  D )  Fn  (
( Base `  D )  X.  ( Base `  D
) ) ) )
104, 9mpbid 202 . . . 4  |-  ( ph  ->  (  Hom  `  D
)  Fn  ( (
Base `  D )  X.  ( Base `  D
) ) )
11 fnov 6117 . . . 4  |-  ( (  Hom  `  D )  Fn  ( ( Base `  D
)  X.  ( Base `  D ) )  <->  (  Hom  `  D )  =  ( x  e.  ( Base `  D ) ,  y  e.  ( Base `  D
)  |->  ( x (  Hom  `  D )
y ) ) )
1210, 11sylib 189 . . 3  |-  ( ph  ->  (  Hom  `  D
)  =  ( x  e.  ( Base `  D
) ,  y  e.  ( Base `  D
)  |->  ( x (  Hom  `  D )
y ) ) )
136, 12eqtrd 2419 . 2  |-  ( ph  ->  H  =  ( x  e.  ( Base `  D
) ,  y  e.  ( Base `  D
)  |->  ( x (  Hom  `  D )
y ) ) )
14 eqid 2387 . . 3  |-  (  Homf  `  D )  =  (  Homf 
`  D )
15 eqid 2387 . . 3  |-  ( Base `  D )  =  (
Base `  D )
16 eqid 2387 . . 3  |-  (  Hom  `  D )  =  (  Hom  `  D )
1714, 15, 16homffval 13844 . 2  |-  (  Homf  `  D )  =  ( x  e.  ( Base `  D ) ,  y  e.  ( Base `  D
)  |->  ( x (  Hom  `  D )
y ) )
1813, 17syl6eqr 2437 1  |-  ( ph  ->  H  =  (  Homf  `  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    C_ wss 3263    X. cxp 4816    Fn wfn 5389   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   Basecbs 13396    Hom chom 13467    Homf chomf 13818    |`cat cresc 13935
This theorem is referenced by:  subsubc  13977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-hom 13480  df-homf 13822  df-resc 13938
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