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Theorem ustbas 22153
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 22127 . . . 4 UnifOn Fn V
2 fnfun 6101 . . . 4 (UnifOn Fn V → Fun UnifOn)
3 elunirn 6624 . . . 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 22151 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = dom 𝑈)
6 ustbas.1 . . . . . . . 8 𝑋 = dom 𝑈
75, 6syl6eqr 2776 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = 𝑋)
87fveq2d 6308 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑥) → (UnifOn‘𝑥) = (UnifOn‘𝑋))
98eleq2d 2789 . . . . 5 (𝑈 ∈ (UnifOn‘𝑥) → (𝑈 ∈ (UnifOn‘𝑥) ↔ 𝑈 ∈ (UnifOn‘𝑋)))
109ibi 256 . . . 4 (𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋))
1110rexlimivw 3131 . . 3 (∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋))
124, 11sylbi 207 . 2 (𝑈 ran UnifOn → 𝑈 ∈ (UnifOn‘𝑋))
13 elrnust 22150 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
1412, 13impbii 199 1 (𝑈 ran UnifOn ↔ 𝑈 ∈ (UnifOn‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1596  wcel 2103  wrex 3015  Vcvv 3304   cuni 4544  dom cdm 5218  ran crn 5219  Fun wfun 5995   Fn wfn 5996  cfv 6001  UnifOncust 22125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-iota 5964  df-fun 6003  df-fn 6004  df-fv 6009  df-ust 22126
This theorem is referenced by: (None)
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