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Theorem homfval 15549
Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
homffval.f  |-  F  =  ( Hom f  `  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  =  ( Hom f  `  C )
2 homffval.b . . . 4  |-  B  =  ( Base `  C
)
3 homffval.h . . . 4  |-  H  =  ( Hom  `  C
)
41, 2, 3homffval 15547 . . 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 6305 . . 3  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x H y )  =  ( X H Y ) )
76adantl 467 . 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 6324 . . 3  |-  ( X H Y )  e. 
_V
1110a1i 11 . 2  |-  ( ph  ->  ( X H Y )  e.  _V )
125, 7, 8, 9, 11ovmpt2d 6429 1  |-  ( ph  ->  ( X F Y )  =  ( X H Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   _Vcvv 3078   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298   Basecbs 15081   Hom chom 15161   Hom f chomf 15524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-homf 15528
This theorem is referenced by:  homfeqval  15554  comfffval2  15558  comffval2  15559  comfval2  15560  catsubcat  15696  subcss2  15700  fullsubc  15707  fullresc  15708  funcres2c  15758  hof1  16091  hofcllem  16095  hofcl  16096  yonffthlem  16119  srhmsubc  38893  srhmsubcALTV  38912
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