Theorem ucnprima 24169

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 24168	. . 3 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊)
7	5	mpofun 7513	. . . . 5 ⊢ Fun 𝐺
8		ustssxp 24092	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
9	1, 8	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
10		opex 5424	. . . . . . 7 ⊢ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ ∈ V
11	5, 10	dmmpo 8050	. . . . . 6 ⊢ dom 𝐺 = (𝑋 × 𝑋)
12	9, 11	sseqtrrdi 3988	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ dom 𝐺)
13		funimass3 7026	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝑟 ⊆ dom 𝐺) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ 𝑟 ⊆ (◡𝐺 “ 𝑊)))
14	7, 12, 13	sylancr 587	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ 𝑟 ⊆ (◡𝐺 “ 𝑊)))
15	14	rexbidva 3155	. . 3 ⊢ (𝜑 → (∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊 ↔ ∃𝑟 ∈ 𝑈 𝑟 ⊆ (◡𝐺 “ 𝑊)))
16	6, 15	mpbid 232	. 2 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 𝑟 ⊆ (◡𝐺 “ 𝑊))
17	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
18		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ∈ 𝑈)
19		cnvimass 6053	. . . . . 6 ⊢ (◡𝐺 “ 𝑊) ⊆ dom 𝐺
20	19, 11	sseqtri 3995	. . . . 5 ⊢ (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋)
21	20	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋))
22		ustssel 24093	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟 ∈ 𝑈 ∧ (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋)) → (𝑟 ⊆ (◡𝐺 “ 𝑊) → (◡𝐺 “ 𝑊) ∈ 𝑈))
23	17, 18, 21, 22	syl3anc 1373	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (𝑟 ⊆ (◡𝐺 “ 𝑊) → (◡𝐺 “ 𝑊) ∈ 𝑈))
24	23	rexlimdva 3134	. 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ⊆ wss 3914  ⟨cop 4595   × cxp 5636  ^◡ccnv 5637  dom cdm 5638   “ cima 5641  Fun wfun 6505  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  UnifOncust 24087   Cn_ucucn 24162

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 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-map 8801  df-ust 24088  df-ucn 24163

This theorem is referenced by:  fmucnd  24179