Theorem homfeqbas 17742

Description: Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypothesis

Ref	Expression
homfeqbas.1	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))

Assertion

Ref	Expression
homfeqbas	⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))

Proof of Theorem homfeqbas

Step	Hyp	Ref	Expression
1		homfeqbas.1	. . . . 5 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
2	1	dmeqd 5886	. . . 4 ⊢ (𝜑 → dom (Homf ‘𝐶) = dom (Homf ‘𝐷))
3		eqid 2765	. . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
4		eqid 2765	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	3, 4	homffn 17739	. . . . 5 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
6	5	fndmi 6629	. . . 4 ⊢ dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶))
7		eqid 2765	. . . . . 6 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷)
8		eqid 2765	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
9	7, 8	homffn 17739	. . . . 5 ⊢ (Homf ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))
10	9	fndmi 6629	. . . 4 ⊢ dom (Homf ‘𝐷) = ((Base‘𝐷) × (Base‘𝐷))
11	2, 6, 10	3eqtr3g 2823	. . 3 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷)))
12	11	dmeqd 5886	. 2 ⊢ (𝜑 → dom ((Base‘𝐶) × (Base‘𝐶)) = dom ((Base‘𝐷) × (Base‘𝐷)))
13		dmxpid 5911	. 2 ⊢ dom ((Base‘𝐶) × (Base‘𝐶)) = (Base‘𝐶)
14		dmxpid 5911	. 2 ⊢ dom ((Base‘𝐷) × (Base‘𝐷)) = (Base‘𝐷)
15	12, 13, 14	3eqtr3g 2823	1 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))

Syntax hints:   → wi 4   = wceq 1563   × cxp 5650  dom cdm 5652  ‘cfv 6525  Basecbs 17259  Hom_f chomf 17712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-homf 17716

This theorem is referenced by:  homfeqval  17743  comfeqd  17753  comfeqval  17754  catpropd  17755  cidpropd  17756  oppccomfpropd  17773  monpropd  17784  funcpropd  17949  fullpropd  17969  fthpropd  17970  natpropd  18026  fucpropd  18027  xpcpropd  18254  curfpropd  18279  hofpropd  18313  sectpropdlem  49665  invpropdlem  49667  isopropdlem  49669  cicpropdlem  49678  idfu1stalem  49729  fthcomf  49786  uppropd  49810  initopropd  49872  termopropd  49873  oppcthinco  50068  thincpropd  50071  termcpropd  50132