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Theorem ustuqtop5 22269
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
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 22230 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
21adantr 466 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
3 snssi 4475 . . . . . . . . 9 (𝑝𝑋 → {𝑝} ⊆ 𝑋)
4 dfss 3738 . . . . . . . . 9 ({𝑝} ⊆ 𝑋 ↔ {𝑝} = ({𝑝} ∩ 𝑋))
53, 4sylib 208 . . . . . . . 8 (𝑝𝑋 → {𝑝} = ({𝑝} ∩ 𝑋))
6 incom 3956 . . . . . . . 8 ({𝑝} ∩ 𝑋) = (𝑋 ∩ {𝑝})
75, 6syl6req 2822 . . . . . . 7 (𝑝𝑋 → (𝑋 ∩ {𝑝}) = {𝑝})
8 snnzg 4444 . . . . . . 7 (𝑝𝑋 → {𝑝} ≠ ∅)
97, 8eqnetrd 3010 . . . . . 6 (𝑝𝑋 → (𝑋 ∩ {𝑝}) ≠ ∅)
109adantl 467 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑋 ∩ {𝑝}) ≠ ∅)
11 xpima2 5718 . . . . 5 ((𝑋 ∩ {𝑝}) ≠ ∅ → ((𝑋 × 𝑋) “ {𝑝}) = 𝑋)
1210, 11syl 17 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ((𝑋 × 𝑋) “ {𝑝}) = 𝑋)
1312eqcomd 2777 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 = ((𝑋 × 𝑋) “ {𝑝}))
14 imaeq1 5601 . . . . 5 (𝑤 = (𝑋 × 𝑋) → (𝑤 “ {𝑝}) = ((𝑋 × 𝑋) “ {𝑝}))
1514eqeq2d 2781 . . . 4 (𝑤 = (𝑋 × 𝑋) → (𝑋 = (𝑤 “ {𝑝}) ↔ 𝑋 = ((𝑋 × 𝑋) “ {𝑝})))
1615rspcev 3460 . . 3 (((𝑋 × 𝑋) ∈ 𝑈𝑋 = ((𝑋 × 𝑋) “ {𝑝})) → ∃𝑤𝑈 𝑋 = (𝑤 “ {𝑝}))
172, 13, 16syl2anc 573 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ∃𝑤𝑈 𝑋 = (𝑤 “ {𝑝}))
18 elfvex 6364 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
19 utopustuq.1 . . . 4 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
2019ustuqtoplem 22263 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑋 ∈ V) → (𝑋 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑋 = (𝑤 “ {𝑝})))
2118, 20mpidan 669 . 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 382   = wceq 1631  wcel 2145  wne 2943  wrex 3062  Vcvv 3351  cin 3722  wss 3723  c0 4063  {csn 4317  cmpt 4864   × cxp 5248  ran crn 5251  cima 5253  cfv 6030  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 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
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 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ust 22224
This theorem is referenced by:  ustuqtop  22270  utopsnneiplem  22271
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