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

Theorem homffval 13594
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 5525 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
3 homffval.b . . . . . 6  |-  B  =  ( Base `  C
)
42, 3syl6eqr 2333 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
5 fveq2 5525 . . . . . . 7  |-  ( c  =  C  ->  (  Hom  `  c )  =  (  Hom  `  C
) )
6 homffval.h . . . . . . 7  |-  H  =  (  Hom  `  C
)
75, 6syl6eqr 2333 . . . . . 6  |-  ( c  =  C  ->  (  Hom  `  c )  =  H )
87oveqd 5875 . . . . 5  |-  ( c  =  C  ->  (
x (  Hom  `  c
) y )  =  ( x H y ) )
94, 4, 8mpt2eq123dv 5910 . . . 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 13572 . . . 4  |-  Homf  =  (
c  e.  _V  |->  ( x  e.  ( Base `  c ) ,  y  e.  ( Base `  c
)  |->  ( x (  Hom  `  c )
y ) ) )
11 fvex 5539 . . . . . 6  |-  ( Base `  C )  e.  _V
123, 11eqeltri 2353 . . . . 5  |-  B  e. 
_V
1312, 12mpt2ex 6198 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )  e.  _V
149, 10, 13fvmpt 5602 . . 3  |-  ( C  e.  _V  ->  (  Homf  `  C )  =  ( x  e.  B , 
y  e.  B  |->  ( x H y ) ) )
15 mpt20 6199 . . . . 5  |-  ( x  e.  (/) ,  y  e.  (/)  |->  ( x H y ) )  =  (/)
1615eqcomi 2287 . . . 4  |-  (/)  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x H y ) )
17 fvprc 5519 . . . 4  |-  ( -.  C  e.  _V  ->  (  Homf 
`  C )  =  (/) )
18 fvprc 5519 . . . . . 6  |-  ( -.  C  e.  _V  ->  (
Base `  C )  =  (/) )
193, 18syl5eq 2327 . . . . 5  |-  ( -.  C  e.  _V  ->  B  =  (/) )
20 mpt2eq12 5908 . . . . 5  |-  ( ( B  =  (/)  /\  B  =  (/) )  ->  (
x  e.  B , 
y  e.  B  |->  ( x H y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x H y ) ) )
2119, 19, 20syl2anc 642 . . . 4  |-  ( -.  C  e.  _V  ->  ( x  e.  B , 
y  e.  B  |->  ( x H y ) )  =  ( x  e.  (/) ,  y  e.  (/)  |->  ( x H y ) ) )
2216, 17, 213eqtr4a 2341 . . 3  |-  ( -.  C  e.  _V  ->  (  Homf 
`  C )  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) ) )
2314, 22pm2.61i 156 . 2  |-  (  Homf  `  C )  =  ( x  e.  B , 
y  e.  B  |->  ( x H y ) )
241, 23eqtri 2303 1  |-  F  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148    Hom chom 13219    Homf chomf 13568
This theorem is referenced by:  homfval  13595  homffn  13596  homfeq  13597  oppchomf  13623  reschomf  13708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-homf 13572
  Copyright terms: Public domain W3C validator