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Theorem homfeqbas 14753
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 5149 . . . 4  |-  ( ph  ->  dom  ( Hom f  `  C )  =  dom  ( Hom f  `  D ) )
3 eqid 2454 . . . . . 6  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
4 eqid 2454 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
53, 4homffn 14750 . . . . 5  |-  ( Hom f  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
6 fndm 5617 . . . . 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 2454 . . . . . 6  |-  ( Hom f  `  D )  =  ( Hom f  `  D )
9 eqid 2454 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
108, 9homffn 14750 . . . . 5  |-  ( Hom f  `  D )  Fn  (
( Base `  D )  X.  ( Base `  D
) )
11 fndm 5617 . . . . 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 2518 . . 3  |-  ( ph  ->  ( ( Base `  C
)  X.  ( Base `  C ) )  =  ( ( Base `  D
)  X.  ( Base `  D ) ) )
1413dmeqd 5149 . 2  |-  ( ph  ->  dom  ( ( Base `  C )  X.  ( Base `  C ) )  =  dom  ( (
Base `  D )  X.  ( Base `  D
) ) )
15 dmxpid 5166 . 2  |-  dom  (
( Base `  C )  X.  ( Base `  C
) )  =  (
Base `  C )
16 dmxpid 5166 . 2  |-  dom  (
( Base `  D )  X.  ( Base `  D
) )  =  (
Base `  D )
1714, 15, 163eqtr3g 2518 1  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    X. cxp 4945   dom cdm 4947    Fn wfn 5520   ` cfv 5525   Basecbs 14291   Hom f chomf 14722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-homf 14726
This theorem is referenced by:  homfeqval  14754  comfeqd  14764  comfeqval  14765  catpropd  14766  cidpropd  14767  oppccomfpropd  14784  monpropd  14794  funcpropd  14928  fullpropd  14948  fthpropd  14949  natpropd  15004  fucpropd  15005  xpcpropd  15136  curfpropd  15161  hofpropd  15195
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