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

Theorem homfeqval 15188
Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
homfeqval.b  |-  B  =  ( Base `  C
)
homfeqval.h  |-  H  =  ( Hom  `  C
)
homfeqval.j  |-  J  =  ( Hom  `  D
)
homfeqval.1  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
homfeqval.x  |-  ( ph  ->  X  e.  B )
homfeqval.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
homfeqval  |-  ( ph  ->  ( X H Y )  =  ( X J Y ) )

Proof of Theorem homfeqval
StepHypRef Expression
1 homfeqval.1 . . 3  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
21oveqd 6287 . 2  |-  ( ph  ->  ( X ( Hom f  `  C ) Y )  =  ( X ( Hom f  `  D ) Y ) )
3 eqid 2454 . . 3  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
4 homfeqval.b . . 3  |-  B  =  ( Base `  C
)
5 homfeqval.h . . 3  |-  H  =  ( Hom  `  C
)
6 homfeqval.x . . 3  |-  ( ph  ->  X  e.  B )
7 homfeqval.y . . 3  |-  ( ph  ->  Y  e.  B )
83, 4, 5, 6, 7homfval 15183 . 2  |-  ( ph  ->  ( X ( Hom f  `  C ) Y )  =  ( X H Y ) )
9 eqid 2454 . . 3  |-  ( Hom f  `  D )  =  ( Hom f  `  D )
10 eqid 2454 . . 3  |-  ( Base `  D )  =  (
Base `  D )
11 homfeqval.j . . 3  |-  J  =  ( Hom  `  D
)
121homfeqbas 15187 . . . . 5  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
134, 12syl5eq 2507 . . . 4  |-  ( ph  ->  B  =  ( Base `  D ) )
146, 13eleqtrd 2544 . . 3  |-  ( ph  ->  X  e.  ( Base `  D ) )
157, 13eleqtrd 2544 . . 3  |-  ( ph  ->  Y  e.  ( Base `  D ) )
169, 10, 11, 14, 15homfval 15183 . 2  |-  ( ph  ->  ( X ( Hom f  `  D ) Y )  =  ( X J Y ) )
172, 8, 163eqtr3d 2503 1  |-  ( ph  ->  ( X H Y )  =  ( X J Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   Basecbs 14719   Hom chom 14798   Hom f chomf 15158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-homf 15162
This theorem is referenced by:  comfeq  15197  comfeqval  15199  catpropd  15200  cidpropd  15201  monpropd  15228  funcpropd  15391  fullpropd  15411  natpropd  15467  xpcpropd  15679  curfpropd  15704  hofpropd  15738
  Copyright terms: Public domain W3C validator