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Theorem ustuqtop0 22264
Description: Lemma for ustuqtop 22270. (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 22252 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈𝑝𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋)
213expa 1111 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) ∧ 𝑝𝑋) → (𝑣 “ {𝑝}) ⊆ 𝑋)
32an32s 631 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑣𝑈) → (𝑣 “ {𝑝}) ⊆ 𝑋)
4 vex 3354 . . . . . . . 8 𝑣 ∈ V
54imaex 7251 . . . . . . 7 (𝑣 “ {𝑝}) ∈ V
65elpw 4303 . . . . . 6 ((𝑣 “ {𝑝}) ∈ 𝒫 𝑋 ↔ (𝑣 “ {𝑝}) ⊆ 𝑋)
73, 6sylibr 224 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑣𝑈) → (𝑣 “ {𝑝}) ∈ 𝒫 𝑋)
87ralrimiva 3115 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ∀𝑣𝑈 (𝑣 “ {𝑝}) ∈ 𝒫 𝑋)
9 eqid 2771 . . . . 5 (𝑣𝑈 ↦ (𝑣 “ {𝑝})) = (𝑣𝑈 ↦ (𝑣 “ {𝑝}))
109rnmptss 6534 . . . 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 6628 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
13 rnexg 7245 . . . . 5 ((𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V → ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) ∈ V)
14 elpwg 4305 . . . . 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 466 . . 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 6527 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  wss 3723  𝒫 cpw 4297  {csn 4316  cmpt 4863  ran crn 5250  cima 5252  wf 6027  cfv 6031  UnifOncust 22223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ust 22224
This theorem is referenced by:  ustuqtop  22270  utopsnneiplem  22271
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