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Theorem homafval 15231
Description: Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homarcl.h  |-  H  =  (Homa
`  C )
homafval.b  |-  B  =  ( Base `  C
)
homafval.c  |-  ( ph  ->  C  e.  Cat )
homafval.j  |-  J  =  ( Hom  `  C
)
Assertion
Ref Expression
homafval  |-  ( ph  ->  H  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  ( J `  x )
) ) )
Distinct variable groups:    x, B    x, C    ph, x
Allowed substitution hints:    H( x)    J( x)

Proof of Theorem homafval
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 homarcl.h . 2  |-  H  =  (Homa
`  C )
2 homafval.c . . 3  |-  ( ph  ->  C  e.  Cat )
3 fveq2 5872 . . . . . . 7  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
4 homafval.b . . . . . . 7  |-  B  =  ( Base `  C
)
53, 4syl6eqr 2526 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  B )
65, 5xpeq12d 5030 . . . . 5  |-  ( c  =  C  ->  (
( Base `  c )  X.  ( Base `  c
) )  =  ( B  X.  B ) )
7 fveq2 5872 . . . . . . . 8  |-  ( c  =  C  ->  ( Hom  `  c )  =  ( Hom  `  C
) )
8 homafval.j . . . . . . . 8  |-  J  =  ( Hom  `  C
)
97, 8syl6eqr 2526 . . . . . . 7  |-  ( c  =  C  ->  ( Hom  `  c )  =  J )
109fveq1d 5874 . . . . . 6  |-  ( c  =  C  ->  (
( Hom  `  c ) `
 x )  =  ( J `  x
) )
1110xpeq2d 5029 . . . . 5  |-  ( c  =  C  ->  ( { x }  X.  ( ( Hom  `  c
) `  x )
)  =  ( { x }  X.  ( J `  x )
) )
126, 11mpteq12dv 4531 . . . 4  |-  ( c  =  C  ->  (
x  e.  ( (
Base `  c )  X.  ( Base `  c
) )  |->  ( { x }  X.  (
( Hom  `  c ) `
 x ) ) )  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  ( J `  x )
) ) )
13 df-homa 15228 . . . 4  |- Homa  =  ( c  e.  Cat  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) )  |->  ( { x }  X.  ( ( Hom  `  c
) `  x )
) ) )
14 fvex 5882 . . . . . . 7  |-  ( Base `  C )  e.  _V
154, 14eqeltri 2551 . . . . . 6  |-  B  e. 
_V
1615, 15xpex 6599 . . . . 5  |-  ( B  X.  B )  e. 
_V
1716mptex 6142 . . . 4  |-  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  ( J `  x )
) )  e.  _V
1812, 13, 17fvmpt 5957 . . 3  |-  ( C  e.  Cat  ->  (Homa `  C
)  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  ( J `  x )
) ) )
192, 18syl 16 . 2  |-  ( ph  ->  (Homa
`  C )  =  ( x  e.  ( B  X.  B ) 
|->  ( { x }  X.  ( J `  x
) ) ) )
201, 19syl5eq 2520 1  |-  ( ph  ->  H  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  ( J `  x )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3118   {csn 4033    |-> cmpt 4511    X. cxp 5003   ` cfv 5594   Basecbs 14507   Hom chom 14583   Catccat 14936  Homachoma 15225
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-homa 15228
This theorem is referenced by:  homaf  15232  homaval  15233
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