Theorem ustuqtop0 24155

Description: Lemma for ustuqtop 24161. (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 24143	. . . . . . . 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 652	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑝}) ⊆ 𝑋)
4		vex 3440	. . . . . . . 8 ⊢ 𝑣 ∈ V
5	4	imaex 7844	. . . . . . 7 ⊢ (𝑣 “ {𝑝}) ∈ V
6	5	elpw 4551	. . . . . 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 3124	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ∀𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ∈ 𝒫 𝑋)
9		eqid 2731	. . . . 5 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝}))
10	9	rnmptss 7056	. . . 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 7155	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
13		rnexg 7832	. . . . 5 ⊢ ((𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
14		elpwg 4550	. . . . 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 7047	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ⊆ wss 3897  𝒫 cpw 4547  {csn 4573   ↦ cmpt 5170  ran crn 5615   “ cima 5617  ⟶wf 6477  ‘cfv 6481  UnifOncust 24115

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ust 24116

This theorem is referenced by:  ustuqtop  24161  utopsnneiplem  24162