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Theorem homafval 14176
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 5720 . . . . . . 7  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
4 homafval.b . . . . . . 7  |-  B  =  ( Base `  C
)
53, 4syl6eqr 2485 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  B )
65, 5xpeq12d 4895 . . . . 5  |-  ( c  =  C  ->  (
( Base `  c )  X.  ( Base `  c
) )  =  ( B  X.  B ) )
7 fveq2 5720 . . . . . . . 8  |-  ( c  =  C  ->  (  Hom  `  c )  =  (  Hom  `  C
) )
8 homafval.j . . . . . . . 8  |-  J  =  (  Hom  `  C
)
97, 8syl6eqr 2485 . . . . . . 7  |-  ( c  =  C  ->  (  Hom  `  c )  =  J )
109fveq1d 5722 . . . . . 6  |-  ( c  =  C  ->  (
(  Hom  `  c ) `
 x )  =  ( J `  x
) )
1110xpeq2d 4894 . . . . 5  |-  ( c  =  C  ->  ( { x }  X.  ( (  Hom  `  c
) `  x )
)  =  ( { x }  X.  ( J `  x )
) )
126, 11mpteq12dv 4279 . . . 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 14173 . . . 4  |- Homa  =  ( c  e.  Cat  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) )  |->  ( { x }  X.  ( (  Hom  `  c
) `  x )
) ) )
14 fvex 5734 . . . . . . 7  |-  ( Base `  C )  e.  _V
154, 14eqeltri 2505 . . . . . 6  |-  B  e. 
_V
1615, 15xpex 4982 . . . . 5  |-  ( B  X.  B )  e. 
_V
1716mptex 5958 . . . 4  |-  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  ( J `  x )
) )  e.  _V
1812, 13, 17fvmpt 5798 . . 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 2479 1  |-  ( ph  ->  H  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  ( J `  x )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806    e. cmpt 4258    X. cxp 4868   ` cfv 5446   Basecbs 13461    Hom chom 13532   Catccat 13881  Homachoma 14170
This theorem is referenced by:  homaf  14177  homaval  14178
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-homa 14173
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