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Theorem homfeqbas 14617
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 5029 . . . 4  |-  ( ph  ->  dom  ( Hom f  `  C )  =  dom  ( Hom f  `  D ) )
3 eqid 2433 . . . . . 6  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
4 eqid 2433 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
53, 4homffn 14614 . . . . 5  |-  ( Hom f  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
6 fndm 5498 . . . . 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 2433 . . . . . 6  |-  ( Hom f  `  D )  =  ( Hom f  `  D )
9 eqid 2433 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
108, 9homffn 14614 . . . . 5  |-  ( Hom f  `  D )  Fn  (
( Base `  D )  X.  ( Base `  D
) )
11 fndm 5498 . . . . 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 2488 . . 3  |-  ( ph  ->  ( ( Base `  C
)  X.  ( Base `  C ) )  =  ( ( Base `  D
)  X.  ( Base `  D ) ) )
1413dmeqd 5029 . 2  |-  ( ph  ->  dom  ( ( Base `  C )  X.  ( Base `  C ) )  =  dom  ( (
Base `  D )  X.  ( Base `  D
) ) )
15 dmxpid 5046 . 2  |-  dom  (
( Base `  C )  X.  ( Base `  C
) )  =  (
Base `  C )
16 dmxpid 5046 . 2  |-  dom  (
( Base `  D )  X.  ( Base `  D
) )  =  (
Base `  D )
1714, 15, 163eqtr3g 2488 1  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1362    X. cxp 4825   dom cdm 4827    Fn wfn 5401   ` cfv 5406   Basecbs 14156   Hom f chomf 14586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-1st 6566  df-2nd 6567  df-homf 14590
This theorem is referenced by:  homfeqval  14618  comfeqd  14628  comfeqval  14629  catpropd  14630  cidpropd  14631  oppccomfpropd  14648  monpropd  14658  funcpropd  14792  fullpropd  14812  fthpropd  14813  natpropd  14868  fucpropd  14869  xpcpropd  15000  curfpropd  15025  hofpropd  15059
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