Theorem homfeqbas 17711

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 5879	. . . 4 ⊢ (𝜑 → dom (Homf ‘𝐶) = dom (Homf ‘𝐷))
3		eqid 2761	. . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
4		eqid 2761	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	3, 4	homffn 17708	. . . . 5 ⊢ (Homf ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
6	5	fndmi 6621	. . . 4 ⊢ dom (Homf ‘𝐶) = ((Base‘𝐶) × (Base‘𝐶))
7		eqid 2761	. . . . . 6 ⊢ (Homf ‘𝐷) = (Homf ‘𝐷)
8		eqid 2761	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
9	7, 8	homffn 17708	. . . . 5 ⊢ (Homf ‘𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))
10	9	fndmi 6621	. . . 4 ⊢ dom (Homf ‘𝐷) = ((Base‘𝐷) × (Base‘𝐷))
11	2, 6, 10	3eqtr3g 2819	. . 3 ⊢ (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷)))
12	11	dmeqd 5879	. 2 ⊢ (𝜑 → dom ((Base‘𝐶) × (Base‘𝐶)) = dom ((Base‘𝐷) × (Base‘𝐷)))
13		dmxpid 5904	. 2 ⊢ dom ((Base‘𝐶) × (Base‘𝐶)) = (Base‘𝐶)
14		dmxpid 5904	. 2 ⊢ dom ((Base‘𝐷) × (Base‘𝐷)) = (Base‘𝐷)
15	12, 13, 14	3eqtr3g 2819	1 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))

Syntax hints:   → wi 4   = wceq 1559   × cxp 5643  dom cdm 5645  ‘cfv 6517  Basecbs 17228  Hom_f chomf 17681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-homf 17685

This theorem is referenced by:  homfeqval  17712  comfeqd  17722  comfeqval  17723  catpropd  17724  cidpropd  17725  oppccomfpropd  17742  monpropd  17753  funcpropd  17918  fullpropd  17938  fthpropd  17939  natpropd  17995  fucpropd  17996  xpcpropd  18223  curfpropd  18248  hofpropd  18282  sectpropdlem  49621  invpropdlem  49623  isopropdlem  49625  cicpropdlem  49634  idfu1stalem  49685  fthcomf  49742  uppropd  49766  initopropd  49828  termopropd  49829  oppcthinco  50024  thincpropd  50027  termcpropd  50088