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Theorem ustuqtop2 23375
Description: Lemma for ustuqtop 23379. (Contributed by Thierry Arnoux, 11-Jan-2018.)
Hypothesis
Ref Expression
utopustuq.1 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
Assertion
Ref Expression
ustuqtop2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
Distinct variable groups:   𝑣,𝑝,𝑈   𝑋,𝑝,𝑣   𝑁,𝑝
Allowed substitution hint:   𝑁(𝑣)

Proof of Theorem ustuqtop2
Dummy variables 𝑤 𝑎 𝑏 𝑢 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp-6l 783 . . . . . . . 8 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋))
2 simp-7l 785 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑈 ∈ (UnifOn‘𝑋))
3 simp-4r 780 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑤𝑈)
4 simplr 765 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑢𝑈)
5 ustincl 23340 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈𝑢𝑈) → (𝑤𝑢) ∈ 𝑈)
62, 3, 4, 5syl3anc 1369 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑤𝑢) ∈ 𝑈)
7 ineq12 4146 . . . . . . . . . . 11 ((𝑎 = (𝑤 “ {𝑝}) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝})))
8 inimasn 6056 . . . . . . . . . . . 12 (𝑝 ∈ V → ((𝑤𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝})))
98elv 3436 . . . . . . . . . . 11 ((𝑤𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝}))
107, 9eqtr4di 2797 . . . . . . . . . 10 ((𝑎 = (𝑤 “ {𝑝}) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) = ((𝑤𝑢) “ {𝑝}))
1110ad4ant24 750 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) = ((𝑤𝑢) “ {𝑝}))
12 imaeq1 5961 . . . . . . . . . 10 (𝑥 = (𝑤𝑢) → (𝑥 “ {𝑝}) = ((𝑤𝑢) “ {𝑝}))
1312rspceeqv 3575 . . . . . . . . 9 (((𝑤𝑢) ∈ 𝑈 ∧ (𝑎𝑏) = ((𝑤𝑢) “ {𝑝})) → ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝}))
146, 11, 13syl2anc 583 . . . . . . . 8 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝}))
15 vex 3434 . . . . . . . . . . 11 𝑎 ∈ V
1615inex1 5244 . . . . . . . . . 10 (𝑎𝑏) ∈ V
17 utopustuq.1 . . . . . . . . . . 11 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
1817ustuqtoplem 23372 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑎𝑏) ∈ V) → ((𝑎𝑏) ∈ (𝑁𝑝) ↔ ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝})))
1916, 18mpan2 687 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ((𝑎𝑏) ∈ (𝑁𝑝) ↔ ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝})))
2019biimpar 477 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝})) → (𝑎𝑏) ∈ (𝑁𝑝))
211, 14, 20syl2anc 583 . . . . . . 7 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) ∈ (𝑁𝑝))
2217ustuqtoplem 23372 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏 ∈ V) → (𝑏 ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝})))
2322elvd 3437 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑏 ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝})))
2423biimpa 476 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏 ∈ (𝑁𝑝)) → ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝}))
2524ad5ant13 753 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝}))
2621, 25r19.29a 3219 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑎𝑏) ∈ (𝑁𝑝))
2717ustuqtoplem 23372 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝})))
2827elvd 3437 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝})))
2928biimpa 476 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝}))
3029adantr 480 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) → ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝}))
3126, 30r19.29a 3219 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) → (𝑎𝑏) ∈ (𝑁𝑝))
3231ralrimiva 3109 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∀𝑏 ∈ (𝑁𝑝)(𝑎𝑏) ∈ (𝑁𝑝))
3332ralrimiva 3109 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ∀𝑎 ∈ (𝑁𝑝)∀𝑏 ∈ (𝑁𝑝)(𝑎𝑏) ∈ (𝑁𝑝))
34 fvex 6781 . . . 4 (𝑁𝑝) ∈ V
35 inficl 9145 . . . 4 ((𝑁𝑝) ∈ V → (∀𝑎 ∈ (𝑁𝑝)∀𝑏 ∈ (𝑁𝑝)(𝑎𝑏) ∈ (𝑁𝑝) ↔ (fi‘(𝑁𝑝)) = (𝑁𝑝)))
3634, 35ax-mp 5 . . 3 (∀𝑎 ∈ (𝑁𝑝)∀𝑏 ∈ (𝑁𝑝)(𝑎𝑏) ∈ (𝑁𝑝) ↔ (fi‘(𝑁𝑝)) = (𝑁𝑝))
3733, 36sylib 217 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) = (𝑁𝑝))
38 eqimss 3981 . 2 ((fi‘(𝑁𝑝)) = (𝑁𝑝) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
3937, 38syl 17 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1541  wcel 2109  wral 3065  wrex 3066  Vcvv 3430  cin 3890  wss 3891  {csn 4566  cmpt 5161  ran crn 5589  cima 5591  cfv 6430  ficfi 9130  UnifOncust 23332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-int 4885  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-om 7701  df-1o 8281  df-er 8472  df-en 8708  df-fin 8711  df-fi 9131  df-ust 23333
This theorem is referenced by:  ustuqtop  23379  utopsnneiplem  23380
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