Theorem ucnprima 22414

Description: The preimage by a uniformly continuous function 𝐹 of an entourage 𝑊 of 𝑌 is an entourage of 𝑋. Note of the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 19-Nov-2017.)

Hypotheses

Ref	Expression
ucnprima.1	⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋))
ucnprima.2	⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))
ucnprima.3	⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉))
ucnprima.4	⊢ (𝜑 → 𝑊 ∈ 𝑉)
ucnprima.5	⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)

Assertion

Ref	Expression
ucnprima	⊢ (𝜑 → (◡𝐺 “ 𝑊) ∈ 𝑈)

Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝑋,𝑦   𝑥,𝐺,𝑦   𝑥,𝑈,𝑦   𝑥,𝑉   𝑥,𝑊,𝑦   𝑥,𝑌   𝜑,𝑥,𝑦

Allowed substitution hints:   𝑉(𝑦)   𝑌(𝑦)

Proof of Theorem ucnprima

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ucnprima.1	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋))
2		ucnprima.2	. . . 4 ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))
3		ucnprima.3	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉))
4		ucnprima.4	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
5		ucnprima.5	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
6	1, 2, 3, 4, 5	ucnima 22413	. . 3 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊)
7	5	mpt2fun 6996	. . . . 5 ⊢ Fun 𝐺
8		ustssxp 22336	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
9	1, 8	sylan 576	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
10		opex 5123	. . . . . . 7 ⊢ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ V
11	5, 10	dmmpt2 7476	. . . . . 6 ⊢ dom 𝐺 = (𝑋 × 𝑋)
12	9, 11	syl6sseqr 3848	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ dom 𝐺)
13		funimass3 6559	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝑟 ⊆ dom 𝐺) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ 𝑟 ⊆ (◡𝐺 “ 𝑊)))
14	7, 12, 13	sylancr 582	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ 𝑟 ⊆ (◡𝐺 “ 𝑊)))
15	14	rexbidva 3230	. . 3 ⊢ (𝜑 → (∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊 ↔ ∃𝑟 ∈ 𝑈 𝑟 ⊆ (◡𝐺 “ 𝑊)))
16	6, 15	mpbid 224	. 2 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 𝑟 ⊆ (◡𝐺 “ 𝑊))
17	1	adantr 473	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
18		simpr 478	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ∈ 𝑈)
19		cnvimass 5702	. . . . . 6 ⊢ (◡𝐺 “ 𝑊) ⊆ dom 𝐺
20	19, 11	sseqtri 3833	. . . . 5 ⊢ (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋)
21	20	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋))
22		ustssel 22337	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟 ∈ 𝑈 ∧ (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋)) → (𝑟 ⊆ (◡𝐺 “ 𝑊) → (◡𝐺 “ 𝑊) ∈ 𝑈))
23	17, 18, 21, 22	syl3anc 1491	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (𝑟 ⊆ (◡𝐺 “ 𝑊) → (◡𝐺 “ 𝑊) ∈ 𝑈))
24	23	rexlimdva 3212	. 2 ⊢ (𝜑 → (∃𝑟 ∈ 𝑈 𝑟 ⊆ (◡𝐺 “ 𝑊) → (◡𝐺 “ 𝑊) ∈ 𝑈))
25	16, 24	mpd 15	1 ⊢ (𝜑 → (◡𝐺 “ 𝑊) ∈ 𝑈)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ∃wrex 3090   ⊆ wss 3769  ⟨cop 4374   × cxp 5310  ^◡ccnv 5311  dom cdm 5312   “ cima 5315  Fun wfun 6095  ‘cfv 6101  (class class class)co 6878   ↦ cmpt2 6880  UnifOncust 22331   Cn_ucucn 22407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-map 8097  df-ust 22332  df-ucn 22408

This theorem is referenced by:  fmucnd  22424