MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  homffval Structured version   Unicode version

Theorem homffval 13918
Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
homffval.f  |-  F  =  (  Homf 
`  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  =  (  Homf 
`  C )
2 fveq2 5729 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
3 homffval.b . . . . . 6  |-  B  =  ( Base `  C
)
42, 3syl6eqr 2487 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
5 fveq2 5729 . . . . . . 7  |-  ( c  =  C  ->  (  Hom  `  c )  =  (  Hom  `  C
) )
6 homffval.h . . . . . . 7  |-  H  =  (  Hom  `  C
)
75, 6syl6eqr 2487 . . . . . 6  |-  ( c  =  C  ->  (  Hom  `  c )  =  H )
87oveqd 6099 . . . . 5  |-  ( c  =  C  ->  (
x (  Hom  `  c
) y )  =  ( x H y ) )
94, 4, 8mpt2eq123dv 6137 . . . 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 13896 . . . 4  |-  Homf  =  (
c  e.  _V  |->  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c
)  |->  ( x (  Hom  `  c )
y ) ) )
11 fvex 5743 . . . . . 6  |-  ( Base `  C )  e.  _V
123, 11eqeltri 2507 . . . . 5  |-  B  e. 
_V
1312, 12mpt2ex 6426 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )  e.  _V
149, 10, 13fvmpt 5807 . . 3  |-  ( C  e.  _V  ->  (  Homf  `  C )  =  ( x  e.  B , 
y  e.  B  |->  ( x H y ) ) )
15 mpt20 6428 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x H y ) )  =  (/)
1615eqcomi 2441 . . . 4  |-  (/)  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x H y ) )
17 fvprc 5723 . . . 4  |-  ( -.  C  e.  _V  ->  (  Homf 
`  C )  =  (/) )
18 fvprc 5723 . . . . . 6  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
193, 18syl5eq 2481 . . . . 5  |-  ( -.  C  e.  _V  ->  B  =  (/) )
20 mpt2eq12 6135 . . . . 5  |-  ( ( B  =  (/)  /\  B  =  (/) )  ->  (
x  e.  B , 
y  e.  B  |->  ( x H y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x H y ) ) )
2119, 19, 20syl2anc 644 . . . 4  |-  ( -.  C  e.  _V  ->  ( x  e.  B , 
y  e.  B  |->  ( x H y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x H y ) ) )
2216, 17, 213eqtr4a 2495 . . 3  |-  ( -.  C  e.  _V  ->  (  Homf 
`  C )  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) ) )
2314, 22pm2.61i 159 . 2  |-  (  Homf  `  C )  =  ( x  e.  B , 
y  e.  B  |->  ( x H y ) )
241, 23eqtri 2457 1  |-  F  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2957   (/)c0 3629   ` cfv 5455  (class class class)co 6082    e. cmpt2 6084   Basecbs 13470    Hom chom 13541    Homf chomf 13892
This theorem is referenced by:  homfval  13919  homffn  13920  homfeq  13921  oppchomf  13947  reschomf  14032
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-homf 13896
  Copyright terms: Public domain W3C validator