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Theorem homfval 13877
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 13876 . . 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 6053 . . 3  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x H y )  =  ( X H Y ) )
76adantl 453 . 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 6069 . . 3  |-  ( X H Y )  e. 
_V
1110a1i 11 . 2  |-  ( ph  ->  ( X H Y )  e.  _V )
125, 7, 8, 9, 11ovmpt2d 6164 1  |-  ( ph  ->  ( X F Y )  =  ( X H Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2920   ` cfv 5417  (class class class)co 6044    e. cmpt2 6046   Basecbs 13428    Hom chom 13499    Homf chomf 13850
This theorem is referenced by:  homfeqval  13882  comfffval2  13886  comffval2  13887  comfval2  13888  subcss2  13999  fullsubc  14006  fullresc  14007  funcres2c  14057  hof1  14310  hofcllem  14314  hofcl  14315  yonffthlem  14338
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-homf 13854
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