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Theorem homfeqbas 17642
Description: Deduce equality of base sets from equality of Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypothesis
Ref Expression
homfeqbas.1 (𝜑 → (Homf𝐶) = (Homf𝐷))
Assertion
Ref Expression
homfeqbas (𝜑 → (Base‘𝐶) = (Base‘𝐷))

Proof of Theorem homfeqbas
StepHypRef Expression
1 homfeqbas.1 . . . . 5 (𝜑 → (Homf𝐶) = (Homf𝐷))
21dmeqd 5905 . . . 4 (𝜑 → dom (Homf𝐶) = dom (Homf𝐷))
3 eqid 2732 . . . . . 6 (Homf𝐶) = (Homf𝐶)
4 eqid 2732 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
53, 4homffn 17639 . . . . 5 (Homf𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))
65fndmi 6653 . . . 4 dom (Homf𝐶) = ((Base‘𝐶) × (Base‘𝐶))
7 eqid 2732 . . . . . 6 (Homf𝐷) = (Homf𝐷)
8 eqid 2732 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
97, 8homffn 17639 . . . . 5 (Homf𝐷) Fn ((Base‘𝐷) × (Base‘𝐷))
109fndmi 6653 . . . 4 dom (Homf𝐷) = ((Base‘𝐷) × (Base‘𝐷))
112, 6, 103eqtr3g 2795 . . 3 (𝜑 → ((Base‘𝐶) × (Base‘𝐶)) = ((Base‘𝐷) × (Base‘𝐷)))
1211dmeqd 5905 . 2 (𝜑 → dom ((Base‘𝐶) × (Base‘𝐶)) = dom ((Base‘𝐷) × (Base‘𝐷)))
13 dmxpid 5929 . 2 dom ((Base‘𝐶) × (Base‘𝐶)) = (Base‘𝐶)
14 dmxpid 5929 . 2 dom ((Base‘𝐷) × (Base‘𝐷)) = (Base‘𝐷)
1512, 13, 143eqtr3g 2795 1 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541   × cxp 5674  dom cdm 5676  cfv 6543  Basecbs 17146  Homf chomf 17612
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-homf 17616
This theorem is referenced by:  homfeqval  17643  comfeqd  17653  comfeqval  17654  catpropd  17655  cidpropd  17656  oppccomfpropd  17675  monpropd  17686  funcpropd  17853  fullpropd  17873  fthpropd  17874  natpropd  17931  fucpropd  17932  xpcpropd  18163  curfpropd  18188  hofpropd  18222
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