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Theorem ustuqtop5 22171
Description: Lemma for ustuqtop 22172. (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 22132 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
21adantr 472 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
3 snssi 4447 . . . . . . . . 9 (𝑝𝑋 → {𝑝} ⊆ 𝑋)
4 dfss 3695 . . . . . . . . 9 ({𝑝} ⊆ 𝑋 ↔ {𝑝} = ({𝑝} ∩ 𝑋))
53, 4sylib 208 . . . . . . . 8 (𝑝𝑋 → {𝑝} = ({𝑝} ∩ 𝑋))
6 incom 3913 . . . . . . . 8 ({𝑝} ∩ 𝑋) = (𝑋 ∩ {𝑝})
75, 6syl6req 2775 . . . . . . 7 (𝑝𝑋 → (𝑋 ∩ {𝑝}) = {𝑝})
8 snnzg 4414 . . . . . . 7 (𝑝𝑋 → {𝑝} ≠ ∅)
97, 8eqnetrd 2963 . . . . . 6 (𝑝𝑋 → (𝑋 ∩ {𝑝}) ≠ ∅)
109adantl 473 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑋 ∩ {𝑝}) ≠ ∅)
11 xpima2 5688 . . . . 5 ((𝑋 ∩ {𝑝}) ≠ ∅ → ((𝑋 × 𝑋) “ {𝑝}) = 𝑋)
1210, 11syl 17 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ((𝑋 × 𝑋) “ {𝑝}) = 𝑋)
1312eqcomd 2730 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 = ((𝑋 × 𝑋) “ {𝑝}))
14 imaeq1 5571 . . . . 5 (𝑤 = (𝑋 × 𝑋) → (𝑤 “ {𝑝}) = ((𝑋 × 𝑋) “ {𝑝}))
1514eqeq2d 2734 . . . 4 (𝑤 = (𝑋 × 𝑋) → (𝑋 = (𝑤 “ {𝑝}) ↔ 𝑋 = ((𝑋 × 𝑋) “ {𝑝})))
1615rspcev 3413 . . 3 (((𝑋 × 𝑋) ∈ 𝑈𝑋 = ((𝑋 × 𝑋) “ {𝑝})) → ∃𝑤𝑈 𝑋 = (𝑤 “ {𝑝}))
172, 13, 16syl2anc 696 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ∃𝑤𝑈 𝑋 = (𝑤 “ {𝑝}))
18 elfvex 6334 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
19 utopustuq.1 . . . 4 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
2019ustuqtoplem 22165 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑋 ∈ V) → (𝑋 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑋 = (𝑤 “ {𝑝})))
2118, 20mpidan 707 . 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 383   = wceq 1596  wcel 2103  wne 2896  wrex 3015  Vcvv 3304  cin 3679  wss 3680  c0 4023  {csn 4285  cmpt 4837   × cxp 5216  ran crn 5219  cima 5221  cfv 6001  UnifOncust 22125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ust 22126
This theorem is referenced by:  ustuqtop  22172  utopsnneiplem  22173
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