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Theorem homffval 14963
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 2526 . . . . 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 2526 . . . . . 6  |-  ( c  =  C  ->  ( Hom  `  c )  =  H )
87oveqd 6312 . . . . 5  |-  ( c  =  C  ->  (
x ( Hom  `  c
) y )  =  ( x H y ) )
94, 4, 8mpt2eq123dv 6354 . . . 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 14941 . . . 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 2551 . . . . 5  |-  B  e. 
_V
1312, 12mpt2ex 6872 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )  e.  _V
149, 10, 13fvmpt 5957 . . 3  |-  ( C  e.  _V  ->  ( Hom f  `  C )  =  ( x  e.  B , 
y  e.  B  |->  ( x H y ) ) )
15 mpt20 6362 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x H y ) )  =  (/)
1615eqcomi 2480 . . . 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 2520 . . . . 5  |-  ( -.  C  e.  _V  ->  B  =  (/) )
20 mpt2eq12 6352 . . . . 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 2534 . . 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 2496 1  |-  F  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   _Vcvv 3118   (/)c0 3790   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   Basecbs 14506   Hom chom 14582   Hom f chomf 14937
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-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-homf 14941
This theorem is referenced by:  homfval  14964  homffn  14965  homfeq  14966  oppchomf  14992  reschomf  15077
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