Theorem ustuqtop0 24559

Description: Lemma for ustuqtop 24565. (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 24547	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈 ∧ 𝑝 ∈ 𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋)
2	1	3expa 1136	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑝 ∈ 𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋)
3	2	an32s 665	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑝}) ⊆ 𝑋)
4		vex 3455	. . . . . . . 8 ⊢ 𝑣 ∈ V
5	4	imaex 7926	. . . . . . 7 ⊢ (𝑣 “ {𝑝}) ∈ V
6	5	elpw 4561	. . . . . 6 ⊢ ((𝑣 “ {𝑝}) ∈ 𝒫 𝑋 ↔ (𝑣 “ {𝑝}) ⊆ 𝑋)
7	3, 6	sylibr 237	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑝}) ∈ 𝒫 𝑋)
8	7	ralrimiva 3155	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ∀𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ∈ 𝒫 𝑋)
9		eqid 2761	. . . . 5 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝}))
10	9	rnmptss 7123	. . . 4 ⊢ (∀𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ∈ 𝒫 𝑋 → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋)
11	8, 10	syl 18	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋)
12		mptexg 7227	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
13		rnexg 7914	. . . . 5 ⊢ ((𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
14		elpwg 4560	. . . . 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 19	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋 ↔ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋))
16	15	adantr 486	. . 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 260	. 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 7114	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   ⊆ wss 3899  𝒫 cpw 4557  {csn 4584   ↦ cmpt 5186  ran crn 5652   “ cima 5654  ⟶wf 6534  ‘cfv 6538  UnifOncust 24519

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ust 24520

This theorem is used by:  ustuqtop  24565  utopsnneiplem  24566