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Theorem homfeqbas 13599
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 4881 . . . 4  |-  ( ph  ->  dom  (  Homf  `  C )  =  dom  (  Homf  `  D ) )
3 eqid 2283 . . . . . 6  |-  (  Homf  `  C )  =  (  Homf 
`  C )
4 eqid 2283 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
53, 4homffn 13596 . . . . 5  |-  (  Homf  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
6 fndm 5343 . . . . 5  |-  ( (  Homf 
`  C )  Fn  ( ( Base `  C
)  X.  ( Base `  C ) )  ->  dom  (  Homf 
`  C )  =  ( ( Base `  C
)  X.  ( Base `  C ) ) )
75, 6ax-mp 8 . . . 4  |-  dom  (  Homf  `  C )  =  ( ( Base `  C
)  X.  ( Base `  C ) )
8 eqid 2283 . . . . . 6  |-  (  Homf  `  D )  =  (  Homf 
`  D )
9 eqid 2283 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
108, 9homffn 13596 . . . . 5  |-  (  Homf  `  D )  Fn  (
( Base `  D )  X.  ( Base `  D
) )
11 fndm 5343 . . . . 5  |-  ( (  Homf 
`  D )  Fn  ( ( Base `  D
)  X.  ( Base `  D ) )  ->  dom  (  Homf 
`  D )  =  ( ( Base `  D
)  X.  ( Base `  D ) ) )
1210, 11ax-mp 8 . . . 4  |-  dom  (  Homf  `  D )  =  ( ( Base `  D
)  X.  ( Base `  D ) )
132, 7, 123eqtr3g 2338 . . 3  |-  ( ph  ->  ( ( Base `  C
)  X.  ( Base `  C ) )  =  ( ( Base `  D
)  X.  ( Base `  D ) ) )
1413dmeqd 4881 . 2  |-  ( ph  ->  dom  ( ( Base `  C )  X.  ( Base `  C ) )  =  dom  ( (
Base `  D )  X.  ( Base `  D
) ) )
15 dmxpid 4898 . 2  |-  dom  (
( Base `  C )  X.  ( Base `  C
) )  =  (
Base `  C )
16 dmxpid 4898 . 2  |-  dom  (
( Base `  D )  X.  ( Base `  D
) )  =  (
Base `  D )
1714, 15, 163eqtr3g 2338 1  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    X. cxp 4687   dom cdm 4689    Fn wfn 5250   ` cfv 5255   Basecbs 13148    Homf chomf 13568
This theorem is referenced by:  homfeqval  13600  comfeqd  13610  comfeqval  13611  catpropd  13612  cidpropd  13613  oppccomfpropd  13630  monpropd  13640  funcpropd  13774  fullpropd  13794  fthpropd  13795  natpropd  13850  fucpropd  13851  xpcpropd  13982  curfpropd  14007  hofpropd  14041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-homf 13572
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