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Theorem homfeqbas 13698
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 4963 . . . 4  |-  ( ph  ->  dom  (  Homf  `  C )  =  dom  (  Homf  `  D ) )
3 eqid 2358 . . . . . 6  |-  (  Homf  `  C )  =  (  Homf 
`  C )
4 eqid 2358 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
53, 4homffn 13695 . . . . 5  |-  (  Homf  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
6 fndm 5425 . . . . 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 2358 . . . . . 6  |-  (  Homf  `  D )  =  (  Homf 
`  D )
9 eqid 2358 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
108, 9homffn 13695 . . . . 5  |-  (  Homf  `  D )  Fn  (
( Base `  D )  X.  ( Base `  D
) )
11 fndm 5425 . . . . 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 2413 . . 3  |-  ( ph  ->  ( ( Base `  C
)  X.  ( Base `  C ) )  =  ( ( Base `  D
)  X.  ( Base `  D ) ) )
1413dmeqd 4963 . 2  |-  ( ph  ->  dom  ( ( Base `  C )  X.  ( Base `  C ) )  =  dom  ( (
Base `  D )  X.  ( Base `  D
) ) )
15 dmxpid 4980 . 2  |-  dom  (
( Base `  C )  X.  ( Base `  C
) )  =  (
Base `  C )
16 dmxpid 4980 . 2  |-  dom  (
( Base `  D )  X.  ( Base `  D
) )  =  (
Base `  D )
1714, 15, 163eqtr3g 2413 1  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    X. cxp 4769   dom cdm 4771    Fn wfn 5332   ` cfv 5337   Basecbs 13245    Homf chomf 13667
This theorem is referenced by:  homfeqval  13699  comfeqd  13709  comfeqval  13710  catpropd  13711  cidpropd  13712  oppccomfpropd  13729  monpropd  13739  funcpropd  13873  fullpropd  13893  fthpropd  13894  natpropd  13949  fucpropd  13950  xpcpropd  14081  curfpropd  14106  hofpropd  14140
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-homf 13671
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