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Theorem homafval 13877
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 5541 . . . . . . 7  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
4 homafval.b . . . . . . 7  |-  B  =  ( Base `  C
)
53, 4syl6eqr 2346 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  B )
65, 5xpeq12d 4730 . . . . 5  |-  ( c  =  C  ->  (
( Base `  c )  X.  ( Base `  c
) )  =  ( B  X.  B ) )
7 fveq2 5541 . . . . . . . 8  |-  ( c  =  C  ->  (  Hom  `  c )  =  (  Hom  `  C
) )
8 homafval.j . . . . . . . 8  |-  J  =  (  Hom  `  C
)
97, 8syl6eqr 2346 . . . . . . 7  |-  ( c  =  C  ->  (  Hom  `  c )  =  J )
109fveq1d 5543 . . . . . 6  |-  ( c  =  C  ->  (
(  Hom  `  c ) `
 x )  =  ( J `  x
) )
1110xpeq2d 4729 . . . . 5  |-  ( c  =  C  ->  ( { x }  X.  ( (  Hom  `  c
) `  x )
)  =  ( { x }  X.  ( J `  x )
) )
126, 11mpteq12dv 4114 . . . 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 13874 . . . 4  |- Homa  =  ( c  e.  Cat  |->  ( x  e.  ( ( Base `  c
)  X.  ( Base `  c ) )  |->  ( { x }  X.  ( (  Hom  `  c
) `  x )
) ) )
14 fvex 5555 . . . . . . 7  |-  ( Base `  C )  e.  _V
154, 14eqeltri 2366 . . . . . 6  |-  B  e. 
_V
1615, 15xpex 4817 . . . . 5  |-  ( B  X.  B )  e. 
_V
1716mptex 5762 . . . 4  |-  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  ( J `  x )
) )  e.  _V
1812, 13, 17fvmpt 5618 . . 3  |-  ( C  e.  Cat  ->  (Homa `  C
)  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  ( J `  x )
) ) )
192, 18syl 15 . 2  |-  ( ph  ->  (Homa
`  C )  =  ( x  e.  ( B  X.  B ) 
|->  ( { x }  X.  ( J `  x
) ) ) )
201, 19syl5eq 2340 1  |-  ( ph  ->  H  =  ( x  e.  ( B  X.  B )  |->  ( { x }  X.  ( J `  x )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653    e. cmpt 4093    X. cxp 4703   ` cfv 5271   Basecbs 13164    Hom chom 13235   Catccat 13582  Homachoma 13871
This theorem is referenced by:  homaf  13878  homaval  13879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-homa 13874
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