Theorem ustuqtop0 24270

Description: Lemma for ustuqtop 24276. (Contributed by Thierry Arnoux, 11-Jan-2018.)

Hypothesis

Ref	Expression
utopustuq.1	⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))

Assertion

Ref	Expression
ustuqtop0	⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)

Distinct variable groups:   𝑣,𝑝,𝑈   𝑋,𝑝,𝑣   𝑁,𝑝

Allowed substitution hint:   𝑁(𝑣)

Proof of Theorem ustuqtop0

Step	Hyp	Ref	Expression
1		ustimasn 24258	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈 ∧ 𝑝 ∈ 𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋)
2	1	3expa 1118	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑝 ∈ 𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋)
3	2	an32s 651	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑝}) ⊆ 𝑋)
4		vex 3492	. . . . . . . 8 ⊢ 𝑣 ∈ V
5	4	imaex 7954	. . . . . . 7 ⊢ (𝑣 “ {𝑝}) ∈ V
6	5	elpw 4626	. . . . . 6 ⊢ ((𝑣 “ {𝑝}) ∈ 𝒫 𝑋 ↔ (𝑣 “ {𝑝}) ⊆ 𝑋)
7	3, 6	sylibr 234	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑝}) ∈ 𝒫 𝑋)
8	7	ralrimiva 3152	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ∀𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ∈ 𝒫 𝑋)
9		eqid 2740	. . . . 5 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝}))
10	9	rnmptss 7157	. . . 4 ⊢ (∀𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ∈ 𝒫 𝑋 → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋)
11	8, 10	syl 17	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋)
12		mptexg 7258	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
13		rnexg 7942	. . . . 5 ⊢ ((𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
14		elpwg 4625	. . . . 5 ⊢ (ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → (ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋 ↔ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋))
15	12, 13, 14	3syl 18	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋 ↔ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋))
16	15	adantr 480	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋 ↔ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋))
17	11, 16	mpbird 257	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋)
18		utopustuq.1	. 2 ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
19	17, 18	fmptd 7148	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622  {csn 4648   ↦ cmpt 5249  ran crn 5701   “ cima 5703  ⟶wf 6569  ‘cfv 6573  UnifOncust 24229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ust 24230

This theorem is referenced by:  ustuqtop  24276  utopsnneiplem  24277