Theorem isusp 21817

Description: The predicate 𝑊 is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)

Hypotheses

Ref	Expression
isusp.1	⊢ 𝐵 = (Base‘𝑊)
isusp.2	⊢ 𝑈 = (UnifSt‘𝑊)
isusp.3	⊢ 𝐽 = (TopOpen‘𝑊)

Assertion

Ref	Expression
isusp	⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))

Proof of Theorem isusp

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3184	. 2 ⊢ (𝑊 ∈ UnifSp → 𝑊 ∈ V)
2		0nep0 4757	. . . . 5 ⊢ ∅ ≠ {∅}
3		isusp.1	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑊)
4		fvprc 6082	. . . . . . . . . . . 12 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
5	3, 4	syl5eq 2655	. . . . . . . . . . 11 ⊢ (¬ 𝑊 ∈ V → 𝐵 = ∅)
6	5	fveq2d 6092	. . . . . . . . . 10 ⊢ (¬ 𝑊 ∈ V → (UnifOn‘𝐵) = (UnifOn‘∅))
7		ust0 21775	. . . . . . . . . 10 ⊢ (UnifOn‘∅) = {{∅}}
8	6, 7	syl6eq 2659	. . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → (UnifOn‘𝐵) = {{∅}})
9	8	eleq2d 2672	. . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 ∈ {{∅}}))
10		isusp.2	. . . . . . . . . 10 ⊢ 𝑈 = (UnifSt‘𝑊)
11		fvex 6098	. . . . . . . . . 10 ⊢ (UnifSt‘𝑊) ∈ V
12	10, 11	eqeltri 2683	. . . . . . . . 9 ⊢ 𝑈 ∈ V
13	12	elsn 4139	. . . . . . . 8 ⊢ (𝑈 ∈ {{∅}} ↔ 𝑈 = {∅})
14	9, 13	syl6bb 274	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 = {∅}))
15		fvprc 6082	. . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
16	10, 15	syl5eq 2655	. . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → 𝑈 = ∅)
17	16	eqeq1d 2611	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (𝑈 = {∅} ↔ ∅ = {∅}))
18	14, 17	bitrd 266	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ = {∅}))
19	18	necon3bbid 2818	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (¬ 𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ ≠ {∅}))
20	2, 19	mpbiri 246	. . . 4 ⊢ (¬ 𝑊 ∈ V → ¬ 𝑈 ∈ (UnifOn‘𝐵))
21	20	con4i 111	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝐵) → 𝑊 ∈ V)
22	21	adantr 479	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)) → 𝑊 ∈ V)
23		fveq2 6088	. . . . . 6 ⊢ (𝑤 = 𝑊 → (UnifSt‘𝑤) = (UnifSt‘𝑊))
24	23, 10	syl6eqr 2661	. . . . 5 ⊢ (𝑤 = 𝑊 → (UnifSt‘𝑤) = 𝑈)
25		fveq2 6088	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
26	25, 3	syl6eqr 2661	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
27	26	fveq2d 6092	. . . . 5 ⊢ (𝑤 = 𝑊 → (UnifOn‘(Base‘𝑤)) = (UnifOn‘𝐵))
28	24, 27	eleq12d 2681	. . . 4 ⊢ (𝑤 = 𝑊 → ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ↔ 𝑈 ∈ (UnifOn‘𝐵)))
29		fveq2 6088	. . . . . 6 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
30		isusp.3	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑊)
31	29, 30	syl6eqr 2661	. . . . 5 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
32	24	fveq2d 6092	. . . . 5 ⊢ (𝑤 = 𝑊 → (unifTop‘(UnifSt‘𝑤)) = (unifTop‘𝑈))
33	31, 32	eqeq12d 2624	. . . 4 ⊢ (𝑤 = 𝑊 → ((TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)) ↔ 𝐽 = (unifTop‘𝑈)))
34	28, 33	anbi12d 742	. . 3 ⊢ (𝑤 = 𝑊 → (((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤))) ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
35		df-usp 21813	. . 3 ⊢ UnifSp = {𝑤 ∣ ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)))}
36	34, 35	elab2g 3321	. 2 ⊢ (𝑊 ∈ V → (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
37	1, 22, 36	pm5.21nii 366	1 ⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))

Syntax hints:  ¬ wn 3   ↔ wb 194   ∧ wa 382   = wceq 1474   ∈ wcel 1976   ≠ wne 2779  Vcvv 3172  ∅c0 3873  {csn 4124  ‘cfv 5790  Basecbs 15641  TopOpenctopn 15851  UnifOncust 21755  unifTopcutop 21786  UnifStcuss 21809  UnifSpcusp 21810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-res 5040  df-iota 5754  df-fun 5792  df-fv 5798  df-ust 21756  df-usp 21813

This theorem is referenced by:  ressust  21820  ressusp  21821  tususp  21828  uspreg  21830  ucncn  21841  neipcfilu  21852  ucnextcn  21860  xmsusp  22125