Theorem homfeqbas 17875

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 2760	. . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
4		eqid 2760	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	3, 4	homffn 17872	. . . . 5 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
6	5	fndmi 6643	. . . 4 ⊢ dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶))
7		eqid 2760	. . . . . 6 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷)
8		eqid 2760	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
9	7, 8	homffn 17872	. . . . 5 ⊢ (Homf ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))
10	9	fndmi 6643	. . . 4 ⊢ dom (Homf ‘𝐷) = ((Base‘𝐷) × (Base‘𝐷))
11	2, 6, 10	3eqtr3g 2818	. . 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 2818	1 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   × cxp 5648  dom cdm 5650  ‘cfv 6538  Basecbs 17392  Hom_f chomf 17845

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-homf 17849

This theorem is used by:  homfeqval  17876  comfeqd  17886  comfeqval  17887  catpropd  17888  cidpropd  17889  oppccomfpropd  17906  monpropd  17917  funcpropd  18082  fullpropd  18102  fthpropd  18103  natpropd  18159  fucpropd  18160  xpcpropd  18387  curfpropd  18412  hofpropd  18446  sectpropdlem  50243  invpropdlem  50245  isopropdlem  50247  cicpropdlem  50256  idfu1stalem  50309  fthcomf  50366  uppropd  50390  initopropd  50458  termopropd  50459  oppcthinco  50667  thincpropd  50670  termcpropd  50731