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Theorem homfeqbas 16836
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 5620 . . . 4 (𝜑 → dom (Homf𝐶) = dom (Homf𝐷))
3 eqid 2772 . . . . . 6 (Homf𝐶) = (Homf𝐶)
4 eqid 2772 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
53, 4homffn 16833 . . . . 5 (Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
6 fndm 6285 . . . . 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 2772 . . . . . 6 (Homf𝐷) = (Homf𝐷)
9 eqid 2772 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
108, 9homffn 16833 . . . . 5 (Homf𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))
11 fndm 6285 . . . . 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 2831 . . 3 (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷)))
1413dmeqd 5620 . 2 (𝜑 → dom ((Base‘𝐶) × (Base‘𝐶)) = dom ((Base‘𝐷) × (Base‘𝐷)))
15 dmxpid 5640 . 2 dom ((Base‘𝐶) × (Base‘𝐶)) = (Base‘𝐶)
16 dmxpid 5640 . 2 dom ((Base‘𝐷) × (Base‘𝐷)) = (Base‘𝐷)
1714, 15, 163eqtr3g 2831 1 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507   × cxp 5401  dom cdm 5403   Fn wfn 6180  cfv 6185  Basecbs 16337  Homf chomf 16807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-1st 7499  df-2nd 7500  df-homf 16811
This theorem is referenced by:  homfeqval  16837  comfeqd  16847  comfeqval  16848  catpropd  16849  cidpropd  16850  oppccomfpropd  16867  monpropd  16877  funcpropd  17040  fullpropd  17060  fthpropd  17061  natpropd  17116  fucpropd  17117  xpcpropd  17328  curfpropd  17353  hofpropd  17387
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