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Theorem homffval 14642
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 5703 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
3 homffval.b . . . . . 6  |-  B  =  ( Base `  C
)
42, 3syl6eqr 2493 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
5 fveq2 5703 . . . . . . 7  |-  ( c  =  C  ->  ( Hom  `  c )  =  ( Hom  `  C
) )
6 homffval.h . . . . . . 7  |-  H  =  ( Hom  `  C
)
75, 6syl6eqr 2493 . . . . . 6  |-  ( c  =  C  ->  ( Hom  `  c )  =  H )
87oveqd 6120 . . . . 5  |-  ( c  =  C  ->  (
x ( Hom  `  c
) y )  =  ( x H y ) )
94, 4, 8mpt2eq123dv 6160 . . . 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 14620 . . . 4  |-  Hom f  =  (
c  e.  _V  |->  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c
)  |->  ( x ( Hom  `  c )
y ) ) )
11 fvex 5713 . . . . . 6  |-  ( Base `  C )  e.  _V
123, 11eqeltri 2513 . . . . 5  |-  B  e. 
_V
1312, 12mpt2ex 6662 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )  e.  _V
149, 10, 13fvmpt 5786 . . 3  |-  ( C  e.  _V  ->  ( Hom f  `  C )  =  ( x  e.  B , 
y  e.  B  |->  ( x H y ) ) )
15 mpt20 6168 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x H y ) )  =  (/)
1615eqcomi 2447 . . . 4  |-  (/)  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x H y ) )
17 fvprc 5697 . . . 4  |-  ( -.  C  e.  _V  ->  ( Hom f  `  C )  =  (/) )
18 fvprc 5697 . . . . . 6  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
193, 18syl5eq 2487 . . . . 5  |-  ( -.  C  e.  _V  ->  B  =  (/) )
20 mpt2eq12 6158 . . . . 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 2501 . . 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 2463 1  |-  F  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1369    e. wcel 1756   _Vcvv 2984   (/)c0 3649   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105   Basecbs 14186   Hom chom 14261   Hom f chomf 14616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-1st 6589  df-2nd 6590  df-homf 14620
This theorem is referenced by:  homfval  14643  homffn  14644  homfeq  14645  oppchomf  14671  reschomf  14756
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