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Theorem homfval 14951
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 14950 . . 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 6294 . . 3  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x H y )  =  ( X H Y ) )
76adantl 466 . 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 6310 . . 3  |-  ( X H Y )  e. 
_V
1110a1i 11 . 2  |-  ( ph  ->  ( X H Y )  e.  _V )
125, 7, 8, 9, 11ovmpt2d 6415 1  |-  ( ph  ->  ( X F Y )  =  ( X H Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   Basecbs 14493   Hom chom 14569   Hom f chomf 14924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-homf 14928
This theorem is referenced by:  homfeqval  14956  comfffval2  14960  comffval2  14961  comfval2  14962  subcss2  15073  fullsubc  15080  fullresc  15081  funcres2c  15131  hof1  15384  hofcllem  15388  hofcl  15389  yonffthlem  15412
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