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Theorem homffval 15106
Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
homffval.f  |-  F  =  ( Hom f  `  C )
homffval.b  |-  B  =  ( Base `  C
)
homffval.h  |-  H  =  ( Hom  `  C
)
Assertion
Ref Expression
homffval  |-  F  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )
Distinct variable groups:    x, y, B    x, C, y    x, H, y
Allowed substitution hints:    F( x, y)

Proof of Theorem homffval
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 homffval.f . 2  |-  F  =  ( Hom f  `  C )
2 fveq2 5872 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
3 homffval.b . . . . . 6  |-  B  =  ( Base `  C
)
42, 3syl6eqr 2516 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
5 fveq2 5872 . . . . . . 7  |-  ( c  =  C  ->  ( Hom  `  c )  =  ( Hom  `  C
) )
6 homffval.h . . . . . . 7  |-  H  =  ( Hom  `  C
)
75, 6syl6eqr 2516 . . . . . 6  |-  ( c  =  C  ->  ( Hom  `  c )  =  H )
87oveqd 6313 . . . . 5  |-  ( c  =  C  ->  (
x ( Hom  `  c
) y )  =  ( x H y ) )
94, 4, 8mpt2eq123dv 6358 . . . 4  |-  ( c  =  C  ->  (
x  e.  ( Base `  c ) ,  y  e.  ( Base `  c
)  |->  ( x ( Hom  `  c )
y ) )  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) ) )
10 df-homf 15087 . . . 4  |-  Hom f  =  (
c  e.  _V  |->  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c
)  |->  ( x ( Hom  `  c )
y ) ) )
11 fvex 5882 . . . . . 6  |-  ( Base `  C )  e.  _V
123, 11eqeltri 2541 . . . . 5  |-  B  e. 
_V
1312, 12mpt2ex 6876 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )  e.  _V
149, 10, 13fvmpt 5956 . . 3  |-  ( C  e.  _V  ->  ( Hom f  `  C )  =  ( x  e.  B , 
y  e.  B  |->  ( x H y ) ) )
15 mpt20 6366 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x H y ) )  =  (/)
1615eqcomi 2470 . . . 4  |-  (/)  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x H y ) )
17 fvprc 5866 . . . 4  |-  ( -.  C  e.  _V  ->  ( Hom f  `  C )  =  (/) )
18 fvprc 5866 . . . . . 6  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
193, 18syl5eq 2510 . . . . 5  |-  ( -.  C  e.  _V  ->  B  =  (/) )
20 mpt2eq12 6356 . . . . 5  |-  ( ( B  =  (/)  /\  B  =  (/) )  ->  (
x  e.  B , 
y  e.  B  |->  ( x H y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x H y ) ) )
2119, 19, 20syl2anc 661 . . . 4  |-  ( -.  C  e.  _V  ->  ( x  e.  B , 
y  e.  B  |->  ( x H y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x H y ) ) )
2216, 17, 213eqtr4a 2524 . . 3  |-  ( -.  C  e.  _V  ->  ( Hom f  `  C )  =  ( x  e.  B , 
y  e.  B  |->  ( x H y ) ) )
2314, 22pm2.61i 164 . 2  |-  ( Hom f  `  C )  =  ( x  e.  B , 
y  e.  B  |->  ( x H y ) )
241, 23eqtri 2486 1  |-  F  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   Basecbs 14644   Hom chom 14723   Hom f chomf 15083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-homf 15087
This theorem is referenced by:  fnhomeqhomf  15107  homfval  15108  homffn  15109  homfeq  15110  oppchomf  15136  reschomf  15247
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