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Theorem homfeqbas 16337
Description: Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypothesis
Ref Expression
homfeqbas.1 (𝜑 → (Homf𝐶) = (Homf𝐷))
Assertion
Ref Expression
homfeqbas (𝜑 → (Base‘𝐶) = (Base‘𝐷))

Proof of Theorem homfeqbas
StepHypRef Expression
1 homfeqbas.1 . . . . 5 (𝜑 → (Homf𝐶) = (Homf𝐷))
21dmeqd 5315 . . . 4 (𝜑 → dom (Homf𝐶) = dom (Homf𝐷))
3 eqid 2620 . . . . . 6 (Homf𝐶) = (Homf𝐶)
4 eqid 2620 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
53, 4homffn 16334 . . . . 5 (Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
6 fndm 5978 . . . . 5 ((Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) → dom (Homf𝐶) = ((Base‘𝐶) × (Base‘𝐶)))
75, 6ax-mp 5 . . . 4 dom (Homf𝐶) = ((Base‘𝐶) × (Base‘𝐶))
8 eqid 2620 . . . . . 6 (Homf𝐷) = (Homf𝐷)
9 eqid 2620 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
108, 9homffn 16334 . . . . 5 (Homf𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))
11 fndm 5978 . . . . 5 ((Homf𝐷) Fn ((Base‘𝐷) × (Base‘𝐷)) → dom (Homf𝐷) = ((Base‘𝐷) × (Base‘𝐷)))
1210, 11ax-mp 5 . . . 4 dom (Homf𝐷) = ((Base‘𝐷) × (Base‘𝐷))
132, 7, 123eqtr3g 2677 . . 3 (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷)))
1413dmeqd 5315 . 2 (𝜑 → dom ((Base‘𝐶) × (Base‘𝐶)) = dom ((Base‘𝐷) × (Base‘𝐷)))
15 dmxpid 5334 . 2 dom ((Base‘𝐶) × (Base‘𝐶)) = (Base‘𝐶)
16 dmxpid 5334 . 2 dom ((Base‘𝐷) × (Base‘𝐷)) = (Base‘𝐷)
1714, 15, 163eqtr3g 2677 1 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481   × cxp 5102  dom cdm 5104   Fn wfn 5871  cfv 5876  Basecbs 15838  Homf chomf 16308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-homf 16312
This theorem is referenced by:  homfeqval  16338  comfeqd  16348  comfeqval  16349  catpropd  16350  cidpropd  16351  oppccomfpropd  16368  monpropd  16378  funcpropd  16541  fullpropd  16561  fthpropd  16562  natpropd  16617  fucpropd  16618  xpcpropd  16829  curfpropd  16854  hofpropd  16888
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