Theorem elutop 22842

Description: Open sets in the topology induced by an uniform structure 𝑈 on 𝑋 (Contributed by Thierry Arnoux, 30-Nov-2017.)

Assertion

Ref	Expression
elutop	⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))

Distinct variable groups:   𝑥,𝑣,𝐴   𝑣,𝑈,𝑥   𝑥,𝑋

Allowed substitution hint:   𝑋(𝑣)

Proof of Theorem elutop

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		utopval 22841	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
2	1	eleq2d 2898	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ 𝐴 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎}))
3		sseq2 3993	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑣 “ {𝑥}) ⊆ 𝑎 ↔ (𝑣 “ {𝑥}) ⊆ 𝐴))
4	3	rexbidv 3297	. . . . 5 ⊢ (𝑎 = 𝐴 → (∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))
5	4	raleqbi1dv 3403	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))
6	5	elrab 3680	. . 3 ⊢ (𝐴 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ↔ (𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))
7	2, 6	syl6bb 289	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))
8		elex 3512	. . . . 5 ⊢ (𝐴 ∈ 𝒫 𝑋 → 𝐴 ∈ V)
9	8	a1i 11	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ 𝒫 𝑋 → 𝐴 ∈ V))
10		elfvex 6703	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
11	10	adantr 483	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ V)
12		simpr 487	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
13	11, 12	ssexd 5228	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
14	13	ex 415	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ⊆ 𝑋 → 𝐴 ∈ V))
15		elpwg 4542	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
16	15	a1i 11	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)))
17	9, 14, 16	pm5.21ndd 383	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
18	17	anbi1d 631	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))
19	7, 18	bitrd 281	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139  {crab 3142  Vcvv 3494   ⊆ wss 3936  𝒫 cpw 4539  {csn 4567   “ cima 5558  ‘cfv 6355  UnifOncust 22808  unifTopcutop 22839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-iota 6314  df-fun 6357  df-fn 6358  df-fv 6363  df-ust 22809  df-utop 22840

This theorem is referenced by:  utoptop  22843  utopbas  22844  restutop  22846  restutopopn  22847  ucncn  22894