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Theorem ustbas 23582
Description: Recover the base of an uniform structure π‘ˆ. βˆͺ ran UnifOn is to UnifOn what Top is to TopOn. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Hypothesis
Ref Expression
ustbas.1 𝑋 = dom βˆͺ π‘ˆ
Assertion
Ref Expression
ustbas (π‘ˆ ∈ βˆͺ ran UnifOn ↔ π‘ˆ ∈ (UnifOnβ€˜π‘‹))

Proof of Theorem ustbas
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 ustfn 23556 . . . 4 UnifOn Fn V
2 fnfun 6603 . . . 4 (UnifOn Fn V β†’ Fun UnifOn)
3 elunirn 7199 . . . 4 (Fun UnifOn β†’ (π‘ˆ ∈ βˆͺ ran UnifOn ↔ βˆƒπ‘₯ ∈ dom UnifOnπ‘ˆ ∈ (UnifOnβ€˜π‘₯)))
41, 2, 3mp2b 10 . . 3 (π‘ˆ ∈ βˆͺ ran UnifOn ↔ βˆƒπ‘₯ ∈ dom UnifOnπ‘ˆ ∈ (UnifOnβ€˜π‘₯))
5 ustbas2 23580 . . . . . . . 8 (π‘ˆ ∈ (UnifOnβ€˜π‘₯) β†’ π‘₯ = dom βˆͺ π‘ˆ)
6 ustbas.1 . . . . . . . 8 𝑋 = dom βˆͺ π‘ˆ
75, 6eqtr4di 2795 . . . . . . 7 (π‘ˆ ∈ (UnifOnβ€˜π‘₯) β†’ π‘₯ = 𝑋)
87fveq2d 6847 . . . . . 6 (π‘ˆ ∈ (UnifOnβ€˜π‘₯) β†’ (UnifOnβ€˜π‘₯) = (UnifOnβ€˜π‘‹))
98eleq2d 2824 . . . . 5 (π‘ˆ ∈ (UnifOnβ€˜π‘₯) β†’ (π‘ˆ ∈ (UnifOnβ€˜π‘₯) ↔ π‘ˆ ∈ (UnifOnβ€˜π‘‹)))
109ibi 267 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘₯) β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
1110rexlimivw 3149 . . 3 (βˆƒπ‘₯ ∈ dom UnifOnπ‘ˆ ∈ (UnifOnβ€˜π‘₯) β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
124, 11sylbi 216 . 2 (π‘ˆ ∈ βˆͺ ran UnifOn β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
13 elfvunirn 6875 . 2 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ π‘ˆ ∈ βˆͺ ran UnifOn)
1412, 13impbii 208 1 (π‘ˆ ∈ βˆͺ ran UnifOn ↔ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3074  Vcvv 3446  βˆͺ cuni 4866  dom cdm 5634  ran crn 5635  Fun wfun 6491   Fn wfn 6492  β€˜cfv 6497  UnifOncust 23554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505  df-ust 23555
This theorem is referenced by: (None)
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