Theorem homfeqbas 17606

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 5851	. . . 4 ⊢ (𝜑 → dom (Homf ‘𝐶) = dom (Homf ‘𝐷))
3		eqid 2733	. . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
4		eqid 2733	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	3, 4	homffn 17603	. . . . 5 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
6	5	fndmi 6592	. . . 4 ⊢ dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶))
7		eqid 2733	. . . . . 6 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷)
8		eqid 2733	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
9	7, 8	homffn 17603	. . . . 5 ⊢ (Homf ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))
10	9	fndmi 6592	. . . 4 ⊢ dom (Homf ‘𝐷) = ((Base‘𝐷) × (Base‘𝐷))
11	2, 6, 10	3eqtr3g 2791	. . 3 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷)))
12	11	dmeqd 5851	. 2 ⊢ (𝜑 → dom ((Base‘𝐶) × (Base‘𝐶)) = dom ((Base‘𝐷) × (Base‘𝐷)))
13		dmxpid 5876	. 2 ⊢ dom ((Base‘𝐶) × (Base‘𝐶)) = (Base‘𝐶)
14		dmxpid 5876	. 2 ⊢ dom ((Base‘𝐷) × (Base‘𝐷)) = (Base‘𝐷)
15	12, 13, 14	3eqtr3g 2791	1 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))

Syntax hints:   → wi 4   = wceq 1541   × cxp 5619  dom cdm 5621  ‘cfv 6488  Basecbs 17124  Hom_f chomf 17576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7357  df-oprab 7358  df-mpo 7359  df-1st 7929  df-2nd 7930  df-homf 17580

This theorem is referenced by:  homfeqval  17607  comfeqd  17617  comfeqval  17618  catpropd  17619  cidpropd  17620  oppccomfpropd  17637  monpropd  17648  funcpropd  17813  fullpropd  17833  fthpropd  17834  natpropd  17890  fucpropd  17891  xpcpropd  18118  curfpropd  18143  hofpropd  18177  sectpropdlem  49164  invpropdlem  49166  isopropdlem  49168  cicpropdlem  49177  idfu1stalem  49228  fthcomf  49285  uppropd  49309  initopropd  49371  termopropd  49372  oppcthinco  49567  thincpropd  49570  termcpropd  49631