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Theorem ustuqtop2 22093
Description: Lemma for ustuqtop 22097. (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 827 . . . . . . . 8 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋))
2 simp-7l 829 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑈 ∈ (UnifOn‘𝑋))
3 simp-4r 824 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑤𝑈)
4 simplr 807 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑢𝑈)
5 ustincl 22058 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈𝑢𝑈) → (𝑤𝑢) ∈ 𝑈)
62, 3, 4, 5syl3anc 1366 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑤𝑢) ∈ 𝑈)
7 simpllr 815 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑎 = (𝑤 “ {𝑝}))
8 ineq12 3842 . . . . . . . . . . 11 ((𝑎 = (𝑤 “ {𝑝}) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝})))
9 vex 3234 . . . . . . . . . . . 12 𝑝 ∈ V
10 inimasn 5585 . . . . . . . . . . . 12 (𝑝 ∈ V → ((𝑤𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝})))
119, 10ax-mp 5 . . . . . . . . . . 11 ((𝑤𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝}))
128, 11syl6eqr 2703 . . . . . . . . . 10 ((𝑎 = (𝑤 “ {𝑝}) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) = ((𝑤𝑢) “ {𝑝}))
137, 12sylancom 702 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) = ((𝑤𝑢) “ {𝑝}))
14 imaeq1 5496 . . . . . . . . . . 11 (𝑥 = (𝑤𝑢) → (𝑥 “ {𝑝}) = ((𝑤𝑢) “ {𝑝}))
1514eqeq2d 2661 . . . . . . . . . 10 (𝑥 = (𝑤𝑢) → ((𝑎𝑏) = (𝑥 “ {𝑝}) ↔ (𝑎𝑏) = ((𝑤𝑢) “ {𝑝})))
1615rspcev 3340 . . . . . . . . 9 (((𝑤𝑢) ∈ 𝑈 ∧ (𝑎𝑏) = ((𝑤𝑢) “ {𝑝})) → ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝}))
176, 13, 16syl2anc 694 . . . . . . . 8 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝}))
18 vex 3234 . . . . . . . . . . 11 𝑎 ∈ V
1918inex1 4832 . . . . . . . . . 10 (𝑎𝑏) ∈ V
20 utopustuq.1 . . . . . . . . . . 11 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
2120ustuqtoplem 22090 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑎𝑏) ∈ V) → ((𝑎𝑏) ∈ (𝑁𝑝) ↔ ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝})))
2219, 21mpan2 707 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ((𝑎𝑏) ∈ (𝑁𝑝) ↔ ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝})))
2322biimpar 501 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝})) → (𝑎𝑏) ∈ (𝑁𝑝))
241, 17, 23syl2anc 694 . . . . . . 7 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) ∈ (𝑁𝑝))
25 simp-4l 823 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋))
26 simpllr 815 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑏 ∈ (𝑁𝑝))
27 vex 3234 . . . . . . . . . 10 𝑏 ∈ V
2820ustuqtoplem 22090 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏 ∈ V) → (𝑏 ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝})))
2927, 28mpan2 707 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑏 ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝})))
3029biimpa 500 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏 ∈ (𝑁𝑝)) → ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝}))
3125, 26, 30syl2anc 694 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝}))
3224, 31r19.29a 3107 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑎𝑏) ∈ (𝑁𝑝))
3320ustuqtoplem 22090 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝})))
3418, 33mpan2 707 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝})))
3534biimpa 500 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝}))
3635adantr 480 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) → ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝}))
3732, 36r19.29a 3107 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) → (𝑎𝑏) ∈ (𝑁𝑝))
3837ralrimiva 2995 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∀𝑏 ∈ (𝑁𝑝)(𝑎𝑏) ∈ (𝑁𝑝))
3938ralrimiva 2995 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ∀𝑎 ∈ (𝑁𝑝)∀𝑏 ∈ (𝑁𝑝)(𝑎𝑏) ∈ (𝑁𝑝))
40 fvex 6239 . . . 4 (𝑁𝑝) ∈ V
41 inficl 8372 . . . 4 ((𝑁𝑝) ∈ V → (∀𝑎 ∈ (𝑁𝑝)∀𝑏 ∈ (𝑁𝑝)(𝑎𝑏) ∈ (𝑁𝑝) ↔ (fi‘(𝑁𝑝)) = (𝑁𝑝)))
4240, 41ax-mp 5 . . 3 (∀𝑎 ∈ (𝑁𝑝)∀𝑏 ∈ (𝑁𝑝)(𝑎𝑏) ∈ (𝑁𝑝) ↔ (fi‘(𝑁𝑝)) = (𝑁𝑝))
4339, 42sylib 208 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) = (𝑁𝑝))
44 eqimss 3690 . 2 ((fi‘(𝑁𝑝)) = (𝑁𝑝) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
4543, 44syl 17 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607  {csn 4210  cmpt 4762  ran crn 5144  cima 5146  cfv 5926  ficfi 8357  UnifOncust 22050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-fin 8001  df-fi 8358  df-ust 22051
This theorem is referenced by:  ustuqtop  22097  utopsnneiplem  22098
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