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Theorem homafval 14104
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 5661 . . . . . . 7  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
4 homafval.b . . . . . . 7  |-  B  =  ( Base `  C
)
53, 4syl6eqr 2430 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  B )
65, 5xpeq12d 4836 . . . . 5  |-  ( c  =  C  ->  (
( Base `  c )  X.  ( Base `  c
) )  =  ( B  X.  B ) )
7 fveq2 5661 . . . . . . . 8  |-  ( c  =  C  ->  (  Hom  `  c )  =  (  Hom  `  C
) )
8 homafval.j . . . . . . . 8  |-  J  =  (  Hom  `  C
)
97, 8syl6eqr 2430 . . . . . . 7  |-  ( c  =  C  ->  (  Hom  `  c )  =  J )
109fveq1d 5663 . . . . . 6  |-  ( c  =  C  ->  (
(  Hom  `  c ) `
 x )  =  ( J `  x
) )
1110xpeq2d 4835 . . . . 5  |-  ( c  =  C  ->  ( { x }  X.  ( (  Hom  `  c
) `  x )
)  =  ( { x }  X.  ( J `  x )
) )
126, 11mpteq12dv 4221 . . . 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 14101 . . . 4  |- Homa  =  ( c  e.  Cat  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) )  |->  ( { x }  X.  ( (  Hom  `  c
) `  x )
) ) )
14 fvex 5675 . . . . . . 7  |-  ( Base `  C )  e.  _V
154, 14eqeltri 2450 . . . . . 6  |-  B  e. 
_V
1615, 15xpex 4923 . . . . 5  |-  ( B  X.  B )  e. 
_V
1716mptex 5898 . . . 4  |-  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  ( J `  x )
) )  e.  _V
1812, 13, 17fvmpt 5738 . . 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 2424 1  |-  ( ph  ->  H  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  ( J `  x )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2892   {csn 3750    e. cmpt 4200    X. cxp 4809   ` cfv 5387   Basecbs 13389    Hom chom 13460   Catccat 13809  Homachoma 14098
This theorem is referenced by:  homaf  14105  homaval  14106
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-homa 14101
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