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Theorem homfeqbas 15102
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 5118 . . . 4  |-  ( ph  ->  dom  ( Hom f  `  C )  =  dom  ( Hom f  `  D ) )
3 eqid 2382 . . . . . 6  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
4 eqid 2382 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
53, 4homffn 15099 . . . . 5  |-  ( Hom f  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
6 fndm 5588 . . . . 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 2382 . . . . . 6  |-  ( Hom f  `  D )  =  ( Hom f  `  D )
9 eqid 2382 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
108, 9homffn 15099 . . . . 5  |-  ( Hom f  `  D )  Fn  (
( Base `  D )  X.  ( Base `  D
) )
11 fndm 5588 . . . . 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 2446 . . 3  |-  ( ph  ->  ( ( Base `  C
)  X.  ( Base `  C ) )  =  ( ( Base `  D
)  X.  ( Base `  D ) ) )
1413dmeqd 5118 . 2  |-  ( ph  ->  dom  ( ( Base `  C )  X.  ( Base `  C ) )  =  dom  ( (
Base `  D )  X.  ( Base `  D
) ) )
15 dmxpid 5135 . 2  |-  dom  (
( Base `  C )  X.  ( Base `  C
) )  =  (
Base `  C )
16 dmxpid 5135 . 2  |-  dom  (
( Base `  D )  X.  ( Base `  D
) )  =  (
Base `  D )
1714, 15, 163eqtr3g 2446 1  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    X. cxp 4911   dom cdm 4913    Fn wfn 5491   ` cfv 5496   Basecbs 14634   Hom f chomf 15073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-homf 15077
This theorem is referenced by:  homfeqval  15103  comfeqd  15113  comfeqval  15114  catpropd  15115  cidpropd  15116  oppccomfpropd  15133  monpropd  15143  funcpropd  15306  fullpropd  15326  fthpropd  15327  natpropd  15382  fucpropd  15383  xpcpropd  15594  curfpropd  15619  hofpropd  15653
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