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Theorem homfval 13923
Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
homffval.f  |-  F  =  (  Homf 
`  C )
homffval.b  |-  B  =  ( Base `  C
)
homffval.h  |-  H  =  (  Hom  `  C
)
homfval.x  |-  ( ph  ->  X  e.  B )
homfval.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
homfval  |-  ( ph  ->  ( X F Y )  =  ( X H Y ) )

Proof of Theorem homfval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 homffval.f . . . 4  |-  F  =  (  Homf 
`  C )
2 homffval.b . . . 4  |-  B  =  ( Base `  C
)
3 homffval.h . . . 4  |-  H  =  (  Hom  `  C
)
41, 2, 3homffval 13922 . . 3  |-  F  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )
54a1i 11 . 2  |-  ( ph  ->  F  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) ) )
6 oveq12 6093 . . 3  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x H y )  =  ( X H Y ) )
76adantl 454 . 2  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x H y )  =  ( X H Y ) )
8 homfval.x . 2  |-  ( ph  ->  X  e.  B )
9 homfval.y . 2  |-  ( ph  ->  Y  e.  B )
10 ovex 6109 . . 3  |-  ( X H Y )  e. 
_V
1110a1i 11 . 2  |-  ( ph  ->  ( X H Y )  e.  _V )
125, 7, 8, 9, 11ovmpt2d 6204 1  |-  ( ph  ->  ( X F Y )  =  ( X H Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   Basecbs 13474    Hom chom 13545    Homf chomf 13896
This theorem is referenced by:  homfeqval  13928  comfffval2  13932  comffval2  13933  comfval2  13934  subcss2  14045  fullsubc  14052  fullresc  14053  funcres2c  14103  hof1  14356  hofcllem  14360  hofcl  14361  yonffthlem  14384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-homf 13900
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