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Theorem homfeqbas 14627
Description: Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypothesis
Ref Expression
homfeqbas.1  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
Assertion
Ref Expression
homfeqbas  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )

Proof of Theorem homfeqbas
StepHypRef Expression
1 homfeqbas.1 . . . . 5  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
21dmeqd 5037 . . . 4  |-  ( ph  ->  dom  ( Hom f  `  C )  =  dom  ( Hom f  `  D ) )
3 eqid 2438 . . . . . 6  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
4 eqid 2438 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
53, 4homffn 14624 . . . . 5  |-  ( Hom f  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
6 fndm 5505 . . . . 5  |-  ( ( Hom f  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )  ->  dom  ( Hom f  `  C )  =  ( ( Base `  C
)  X.  ( Base `  C ) ) )
75, 6ax-mp 5 . . . 4  |-  dom  ( Hom f  `  C )  =  ( ( Base `  C
)  X.  ( Base `  C ) )
8 eqid 2438 . . . . . 6  |-  ( Hom f  `  D )  =  ( Hom f  `  D )
9 eqid 2438 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
108, 9homffn 14624 . . . . 5  |-  ( Hom f  `  D )  Fn  (
( Base `  D )  X.  ( Base `  D
) )
11 fndm 5505 . . . . 5  |-  ( ( Hom f  `  D )  Fn  (
( Base `  D )  X.  ( Base `  D
) )  ->  dom  ( Hom f  `  D )  =  ( ( Base `  D
)  X.  ( Base `  D ) ) )
1210, 11ax-mp 5 . . . 4  |-  dom  ( Hom f  `  D )  =  ( ( Base `  D
)  X.  ( Base `  D ) )
132, 7, 123eqtr3g 2493 . . 3  |-  ( ph  ->  ( ( Base `  C
)  X.  ( Base `  C ) )  =  ( ( Base `  D
)  X.  ( Base `  D ) ) )
1413dmeqd 5037 . 2  |-  ( ph  ->  dom  ( ( Base `  C )  X.  ( Base `  C ) )  =  dom  ( (
Base `  D )  X.  ( Base `  D
) ) )
15 dmxpid 5054 . 2  |-  dom  (
( Base `  C )  X.  ( Base `  C
) )  =  (
Base `  C )
16 dmxpid 5054 . 2  |-  dom  (
( Base `  D )  X.  ( Base `  D
) )  =  (
Base `  D )
1714, 15, 163eqtr3g 2493 1  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    X. cxp 4833   dom cdm 4835    Fn wfn 5408   ` cfv 5413   Basecbs 14166   Hom f chomf 14596
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-homf 14600
This theorem is referenced by:  homfeqval  14628  comfeqd  14638  comfeqval  14639  catpropd  14640  cidpropd  14641  oppccomfpropd  14658  monpropd  14668  funcpropd  14802  fullpropd  14822  fthpropd  14823  natpropd  14878  fucpropd  14879  xpcpropd  15010  curfpropd  15035  hofpropd  15069
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