MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustbas Structured version   Visualization version   GIF version

Theorem ustbas 24237
Description: Recover the base of an uniform structure 𝑈. ran UnifOn is to UnifOn what Top is to TopOn. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Hypothesis
Ref Expression
ustbas.1 𝑋 = dom 𝑈
Assertion
Ref Expression
ustbas (𝑈 ran UnifOn ↔ 𝑈 ∈ (UnifOn‘𝑋))

Proof of Theorem ustbas
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ustfn 24211 . . . 4 UnifOn Fn V
2 fnfun 6667 . . . 4 (UnifOn Fn V → Fun UnifOn)
3 elunirn 7272 . . . 4 (Fun UnifOn → (𝑈 ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥)))
41, 2, 3mp2b 10 . . 3 (𝑈 ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))
5 ustbas2 24235 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = dom 𝑈)
6 ustbas.1 . . . . . . . 8 𝑋 = dom 𝑈
75, 6eqtr4di 2794 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = 𝑋)
87fveq2d 6909 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑥) → (UnifOn‘𝑥) = (UnifOn‘𝑋))
98eleq2d 2826 . . . . 5 (𝑈 ∈ (UnifOn‘𝑥) → (𝑈 ∈ (UnifOn‘𝑥) ↔ 𝑈 ∈ (UnifOn‘𝑋)))
109ibi 267 . . . 4 (𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋))
1110rexlimivw 3150 . . 3 (∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋))
124, 11sylbi 217 . 2 (𝑈 ran UnifOn → 𝑈 ∈ (UnifOn‘𝑋))
13 elfvunirn 6937 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
1412, 13impbii 209 1 (𝑈 ran UnifOn ↔ 𝑈 ∈ (UnifOn‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1539  wcel 2107  wrex 3069  Vcvv 3479   cuni 4906  dom cdm 5684  ran crn 5685  Fun wfun 6554   Fn wfn 6555  cfv 6560  UnifOncust 24209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-ust 24210
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator