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Theorem ustuqtop0 21967
Description: Lemma for ustuqtop 21973. (Contributed by Thierry Arnoux, 11-Jan-2018.)
Hypothesis
Ref Expression
utopustuq.1 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (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
StepHypRef Expression
1 ustimasn 21955 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈𝑝𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋)
213expa 1262 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) ∧ 𝑝𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋)
32an32s 845 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑣𝑈) → (𝑣 “ {𝑝}) ⊆ 𝑋)
4 vex 3192 . . . . . . . 8 𝑣 ∈ V
54imaex 7058 . . . . . . 7 (𝑣 “ {𝑝}) ∈ V
65elpw 4141 . . . . . 6 ((𝑣 “ {𝑝}) ∈ 𝒫 𝑋 ↔ (𝑣 “ {𝑝}) ⊆ 𝑋)
73, 6sylibr 224 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑣𝑈) → (𝑣 “ {𝑝}) ∈ 𝒫 𝑋)
87ralrimiva 2961 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ∀𝑣𝑈 (𝑣 “ {𝑝}) ∈ 𝒫 𝑋)
9 eqid 2621 . . . . 5 (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑝}))
109rnmptss 6353 . . . 4 (∀𝑣𝑈 (𝑣 “ {𝑝}) ∈ 𝒫 𝑋 → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋)
118, 10syl 17 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋)
12 mptexg 6444 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
13 rnexg 7052 . . . . 5 ((𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
14 elpwg 4143 . . . . 5 (ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → (ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋 ↔ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋))
1512, 13, 143syl 18 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋 ↔ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋))
1615adantr 481 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋 ↔ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ⊆ 𝒫 𝑋))
1711, 16mpbird 247 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ 𝒫 𝒫 𝑋)
18 utopustuq.1 . 2 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1917, 18fmptd 6346 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3189  wss 3559  𝒫 cpw 4135  {csn 4153  cmpt 4678  ran crn 5080  cima 5082  wf 5848  cfv 5852  UnifOncust 21926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ust 21927
This theorem is referenced by:  ustuqtop  21973  utopsnneiplem  21974
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