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Theorem homfeqbas 15073
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 5195 . . . 4  |-  ( ph  ->  dom  ( Hom f  `  C )  =  dom  ( Hom f  `  D ) )
3 eqid 2443 . . . . . 6  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
4 eqid 2443 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
53, 4homffn 15070 . . . . 5  |-  ( Hom f  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
6 fndm 5670 . . . . 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 2443 . . . . . 6  |-  ( Hom f  `  D )  =  ( Hom f  `  D )
9 eqid 2443 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
108, 9homffn 15070 . . . . 5  |-  ( Hom f  `  D )  Fn  (
( Base `  D )  X.  ( Base `  D
) )
11 fndm 5670 . . . . 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 2507 . . 3  |-  ( ph  ->  ( ( Base `  C
)  X.  ( Base `  C ) )  =  ( ( Base `  D
)  X.  ( Base `  D ) ) )
1413dmeqd 5195 . 2  |-  ( ph  ->  dom  ( ( Base `  C )  X.  ( Base `  C ) )  =  dom  ( (
Base `  D )  X.  ( Base `  D
) ) )
15 dmxpid 5212 . 2  |-  dom  (
( Base `  C )  X.  ( Base `  C
) )  =  (
Base `  C )
16 dmxpid 5212 . 2  |-  dom  (
( Base `  D )  X.  ( Base `  D
) )  =  (
Base `  D )
1714, 15, 163eqtr3g 2507 1  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    X. cxp 4987   dom cdm 4989    Fn wfn 5573   ` cfv 5578   Basecbs 14614   Hom f chomf 15045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-homf 15049
This theorem is referenced by:  homfeqval  15074  comfeqd  15084  comfeqval  15085  catpropd  15086  cidpropd  15087  oppccomfpropd  15104  monpropd  15114  funcpropd  15248  fullpropd  15268  fthpropd  15269  natpropd  15324  fucpropd  15325  xpcpropd  15456  curfpropd  15481  hofpropd  15515
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