Theorem ustinvel 24104

Description: If 𝑉 is an entourage, so is its inverse. Condition U_II of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)

Assertion

Ref	Expression
ustinvel	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ◡𝑉 ∈ 𝑈)

Proof of Theorem ustinvel

Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6899	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
2		isust 24098	. . . . . . 7 ⊢ (𝑋 ∈ V → (𝑈 ∈ (UnifOn‘𝑋) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣)))))
3	1, 2	syl 17	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑈 ∈ (UnifOn‘𝑋) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣)))))
4	3	ibi 267	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣))))
5	4	simp3d 1144	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣)))
6		sseq1 3975	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (𝑣 ⊆ 𝑤 ↔ 𝑉 ⊆ 𝑤))
7	6	imbi1d 341	. . . . . . 7 ⊢ (𝑣 = 𝑉 → ((𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ↔ (𝑉 ⊆ 𝑤 → 𝑤 ∈ 𝑈)))
8	7	ralbidv 3157	. . . . . 6 ⊢ (𝑣 = 𝑉 → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ↔ ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉 ⊆ 𝑤 → 𝑤 ∈ 𝑈)))
9		ineq1 4179	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (𝑣 ∩ 𝑤) = (𝑉 ∩ 𝑤))
10	9	eleq1d 2814	. . . . . . 7 ⊢ (𝑣 = 𝑉 → ((𝑣 ∩ 𝑤) ∈ 𝑈 ↔ (𝑉 ∩ 𝑤) ∈ 𝑈))
11	10	ralbidv 3157	. . . . . 6 ⊢ (𝑣 = 𝑉 → (∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ↔ ∀𝑤 ∈ 𝑈 (𝑉 ∩ 𝑤) ∈ 𝑈))
12		sseq2 3976	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (( I ↾ 𝑋) ⊆ 𝑣 ↔ ( I ↾ 𝑋) ⊆ 𝑉))
13		cnveq 5840	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → ◡𝑣 = ◡𝑉)
14	13	eleq1d 2814	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (◡𝑣 ∈ 𝑈 ↔ ◡𝑉 ∈ 𝑈))
15		sseq2 3976	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → ((𝑤 ∘ 𝑤) ⊆ 𝑣 ↔ (𝑤 ∘ 𝑤) ⊆ 𝑉))
16	15	rexbidv 3158	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣 ↔ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑉))
17	12, 14, 16	3anbi123d 1438	. . . . . 6 ⊢ (𝑣 = 𝑉 → ((( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣) ↔ (( I ↾ 𝑋) ⊆ 𝑉 ∧ ◡𝑉 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑉)))
18	8, 11, 17	3anbi123d 1438	. . . . 5 ⊢ (𝑣 = 𝑉 → ((∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣)) ↔ (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑉 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑉 ∧ ◡𝑉 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑉))))
19	18	rspcv 3587	. . . 4 ⊢ (𝑉 ∈ 𝑈 → (∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣)) → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑉 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑉 ∧ ◡𝑉 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑉))))
20	5, 19	mpan9 506	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑉 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑉 ∧ ◡𝑉 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑉)))
21	20	simp3d 1144	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → (( I ↾ 𝑋) ⊆ 𝑉 ∧ ◡𝑉 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑉))
22	21	simp2d 1143	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ◡𝑉 ∈ 𝑈)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ∩ cin 3916   ⊆ wss 3917  𝒫 cpw 4566   I cid 5535   × cxp 5639  ^◡ccnv 5640   ↾ cres 5643   ∘ ccom 5645  ‘cfv 6514  UnifOncust 24094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-res 5653  df-iota 6467  df-fun 6516  df-fv 6522  df-ust 24095

This theorem is referenced by:  ustexsym  24110  trust  24124