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Theorem homfeqbas 13927
Description: Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypothesis
Ref Expression
homfeqbas.1  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
Assertion
Ref Expression
homfeqbas  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )

Proof of Theorem homfeqbas
StepHypRef Expression
1 homfeqbas.1 . . . . 5  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
21dmeqd 5075 . . . 4  |-  ( ph  ->  dom  (  Homf  `  C )  =  dom  (  Homf  `  D ) )
3 eqid 2438 . . . . . 6  |-  (  Homf  `  C )  =  (  Homf 
`  C )
4 eqid 2438 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
53, 4homffn 13924 . . . . 5  |-  (  Homf  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
6 fndm 5547 . . . . 5  |-  ( (  Homf 
`  C )  Fn  ( ( Base `  C
)  X.  ( Base `  C ) )  ->  dom  (  Homf 
`  C )  =  ( ( Base `  C
)  X.  ( Base `  C ) ) )
75, 6ax-mp 5 . . . 4  |-  dom  (  Homf  `  C )  =  ( ( Base `  C
)  X.  ( Base `  C ) )
8 eqid 2438 . . . . . 6  |-  (  Homf  `  D )  =  (  Homf 
`  D )
9 eqid 2438 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
108, 9homffn 13924 . . . . 5  |-  (  Homf  `  D )  Fn  (
( Base `  D )  X.  ( Base `  D
) )
11 fndm 5547 . . . . 5  |-  ( (  Homf 
`  D )  Fn  ( ( Base `  D
)  X.  ( Base `  D ) )  ->  dom  (  Homf 
`  D )  =  ( ( Base `  D
)  X.  ( Base `  D ) ) )
1210, 11ax-mp 5 . . . 4  |-  dom  (  Homf  `  D )  =  ( ( Base `  D
)  X.  ( Base `  D ) )
132, 7, 123eqtr3g 2493 . . 3  |-  ( ph  ->  ( ( Base `  C
)  X.  ( Base `  C ) )  =  ( ( Base `  D
)  X.  ( Base `  D ) ) )
1413dmeqd 5075 . 2  |-  ( ph  ->  dom  ( ( Base `  C )  X.  ( Base `  C ) )  =  dom  ( (
Base `  D )  X.  ( Base `  D
) ) )
15 dmxpid 5092 . 2  |-  dom  (
( Base `  C )  X.  ( Base `  C
) )  =  (
Base `  C )
16 dmxpid 5092 . 2  |-  dom  (
( Base `  D )  X.  ( Base `  D
) )  =  (
Base `  D )
1714, 15, 163eqtr3g 2493 1  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    X. cxp 4879   dom cdm 4881    Fn wfn 5452   ` cfv 5457   Basecbs 13474    Homf chomf 13896
This theorem is referenced by:  homfeqval  13928  comfeqd  13938  comfeqval  13939  catpropd  13940  cidpropd  13941  oppccomfpropd  13958  monpropd  13968  funcpropd  14102  fullpropd  14122  fthpropd  14123  natpropd  14178  fucpropd  14179  xpcpropd  14310  curfpropd  14335  hofpropd  14369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-homf 13900
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