Theorem ustbas 23379

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.

Step	Hyp	Ref	Expression
1		ustfn 23353	. . . 4 ⊢ UnifOn Fn V
2		fnfun 6533	. . . 4 ⊢ (UnifOn Fn V → Fun UnifOn)
3		elunirn 7124	. . . 4 ⊢ (Fun UnifOn → (𝑈 ∈ ∪ ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥)))
4	1, 2, 3	mp2b 10	. . 3 ⊢ (𝑈 ∈ ∪ ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥))
5		ustbas2 23377	. . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = dom ∪ 𝑈)
6		ustbas.1	. . . . . . . 8 ⊢ 𝑋 = dom ∪ 𝑈
7	5, 6	eqtr4di 2796	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → 𝑥 = 𝑋)
8	7	fveq2d 6778	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → (UnifOn‘𝑥) = (UnifOn‘𝑋))
9	8	eleq2d 2824	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → (𝑈 ∈ (UnifOn‘𝑥) ↔ 𝑈 ∈ (UnifOn‘𝑋)))
10	9	ibi 266	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋))
11	10	rexlimivw 3211	. . 3 ⊢ (∃𝑥 ∈ dom UnifOn𝑈 ∈ (UnifOn‘𝑥) → 𝑈 ∈ (UnifOn‘𝑋))
12	4, 11	sylbi 216	. 2 ⊢ (𝑈 ∈ ∪ ran UnifOn → 𝑈 ∈ (UnifOn‘𝑋))
13		elrnust 23376	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)
14	12, 13	impbii 208	1 ⊢ (𝑈 ∈ ∪ ran UnifOn ↔ 𝑈 ∈ (UnifOn‘𝑋))

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  Vcvv 3432  ∪ cuni 4839  dom cdm 5589  ran crn 5590  Fun wfun 6427   Fn wfn 6428  ‘cfv 6433  UnifOncust 23351

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-ust 23352

This theorem is referenced by: (None)