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Theorem ustuqtop5 21959
Description: Lemma for ustuqtop 21960. (Contributed by Thierry Arnoux, 11-Jan-2018.)
Hypothesis
Ref Expression
utopustuq.1 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
Assertion
Ref Expression
ustuqtop5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
Distinct variable groups:   𝑣,𝑝,𝑈   𝑋,𝑝,𝑣   𝑁,𝑝
Allowed substitution hint:   𝑁(𝑣)

Proof of Theorem ustuqtop5
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ustbasel 21920 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
21adantr 481 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
3 snssi 4308 . . . . . . . . 9 (𝑝𝑋 → {𝑝} ⊆ 𝑋)
4 dfss 3570 . . . . . . . . 9 ({𝑝} ⊆ 𝑋 ↔ {𝑝} = ({𝑝} ∩ 𝑋))
53, 4sylib 208 . . . . . . . 8 (𝑝𝑋 → {𝑝} = ({𝑝} ∩ 𝑋))
6 incom 3783 . . . . . . . 8 ({𝑝} ∩ 𝑋) = (𝑋 ∩ {𝑝})
75, 6syl6req 2672 . . . . . . 7 (𝑝𝑋 → (𝑋 ∩ {𝑝}) = {𝑝})
8 snnzg 4278 . . . . . . 7 (𝑝𝑋 → {𝑝} ≠ ∅)
97, 8eqnetrd 2857 . . . . . 6 (𝑝𝑋 → (𝑋 ∩ {𝑝}) ≠ ∅)
109adantl 482 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑋 ∩ {𝑝}) ≠ ∅)
11 xpima2 5537 . . . . 5 ((𝑋 ∩ {𝑝}) ≠ ∅ → ((𝑋 × 𝑋) “ {𝑝}) = 𝑋)
1210, 11syl 17 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ((𝑋 × 𝑋) “ {𝑝}) = 𝑋)
1312eqcomd 2627 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 = ((𝑋 × 𝑋) “ {𝑝}))
14 imaeq1 5420 . . . . 5 (𝑤 = (𝑋 × 𝑋) → (𝑤 “ {𝑝}) = ((𝑋 × 𝑋) “ {𝑝}))
1514eqeq2d 2631 . . . 4 (𝑤 = (𝑋 × 𝑋) → (𝑋 = (𝑤 “ {𝑝}) ↔ 𝑋 = ((𝑋 × 𝑋) “ {𝑝})))
1615rspcev 3295 . . 3 (((𝑋 × 𝑋) ∈ 𝑈𝑋 = ((𝑋 × 𝑋) “ {𝑝})) → ∃𝑤𝑈 𝑋 = (𝑤 “ {𝑝}))
172, 13, 16syl2anc 692 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ∃𝑤𝑈 𝑋 = (𝑤 “ {𝑝}))
18 elfvex 6178 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
19 utopustuq.1 . . . 4 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
2019ustuqtoplem 21953 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑋 ∈ V) → (𝑋 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑋 = (𝑤 “ {𝑝})))
2118, 20mpidan 703 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑋 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑋 = (𝑤 “ {𝑝})))
2217, 21mpbird 247 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wrex 2908  Vcvv 3186  cin 3554  wss 3555  c0 3891  {csn 4148  cmpt 4673   × cxp 5072  ran crn 5075  cima 5077  cfv 5847  UnifOncust 21913
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ust 21914
This theorem is referenced by:  ustuqtop  21960  utopsnneiplem  21961
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