Theorem homfeqbas 17631

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 5862	. . . 4 ⊢ (𝜑 → dom (Homf ‘𝐶) = dom (Homf ‘𝐷))
3		eqid 2737	. . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
4		eqid 2737	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	3, 4	homffn 17628	. . . . 5 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
6	5	fndmi 6604	. . . 4 ⊢ dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶))
7		eqid 2737	. . . . . 6 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷)
8		eqid 2737	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
9	7, 8	homffn 17628	. . . . 5 ⊢ (Homf ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))
10	9	fndmi 6604	. . . 4 ⊢ dom (Homf ‘𝐷) = ((Base‘𝐷) × (Base‘𝐷))
11	2, 6, 10	3eqtr3g 2795	. . 3 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷)))
12	11	dmeqd 5862	. 2 ⊢ (𝜑 → dom ((Base‘𝐶) × (Base‘𝐶)) = dom ((Base‘𝐷) × (Base‘𝐷)))
13		dmxpid 5887	. 2 ⊢ dom ((Base‘𝐶) × (Base‘𝐶)) = (Base‘𝐶)
14		dmxpid 5887	. 2 ⊢ dom ((Base‘𝐷) × (Base‘𝐷)) = (Base‘𝐷)
15	12, 13, 14	3eqtr3g 2795	1 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))

Syntax hints:   → wi 4   = wceq 1542   × cxp 5630  dom cdm 5632  ‘cfv 6500  Basecbs 17148  Hom_f chomf 17601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-homf 17605

This theorem is referenced by:  homfeqval  17632  comfeqd  17642  comfeqval  17643  catpropd  17644  cidpropd  17645  oppccomfpropd  17662  monpropd  17673  funcpropd  17838  fullpropd  17858  fthpropd  17859  natpropd  17915  fucpropd  17916  xpcpropd  18143  curfpropd  18168  hofpropd  18202  sectpropdlem  49392  invpropdlem  49394  isopropdlem  49396  cicpropdlem  49405  idfu1stalem  49456  fthcomf  49513  uppropd  49537  initopropd  49599  termopropd  49600  oppcthinco  49795  thincpropd  49798  termcpropd  49859