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Theorem ustuqtop2 21951
Description: Lemma for ustuqtop 21955. (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 809 . . . . . . . 8 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋))
2 simp-7l 811 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑈 ∈ (UnifOn‘𝑋))
3 simp-4r 806 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑤𝑈)
4 simplr 791 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑢𝑈)
5 ustincl 21916 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈𝑢𝑈) → (𝑤𝑢) ∈ 𝑈)
62, 3, 4, 5syl3anc 1323 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑤𝑢) ∈ 𝑈)
7 simpllr 798 . . . . . . . . . 10 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑎 = (𝑤 “ {𝑝}))
8 ineq12 3792 . . . . . . . . . . 11 ((𝑎 = (𝑤 “ {𝑝}) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝})))
9 vex 3194 . . . . . . . . . . . 12 𝑝 ∈ V
10 inimasn 5513 . . . . . . . . . . . 12 (𝑝 ∈ V → ((𝑤𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝})))
119, 10ax-mp 5 . . . . . . . . . . 11 ((𝑤𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝}))
128, 11syl6eqr 2678 . . . . . . . . . 10 ((𝑎 = (𝑤 “ {𝑝}) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) = ((𝑤𝑢) “ {𝑝}))
137, 12sylancom 700 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) = ((𝑤𝑢) “ {𝑝}))
14 imaeq1 5424 . . . . . . . . . . 11 (𝑥 = (𝑤𝑢) → (𝑥 “ {𝑝}) = ((𝑤𝑢) “ {𝑝}))
1514eqeq2d 2636 . . . . . . . . . 10 (𝑥 = (𝑤𝑢) → ((𝑎𝑏) = (𝑥 “ {𝑝}) ↔ (𝑎𝑏) = ((𝑤𝑢) “ {𝑝})))
1615rspcev 3300 . . . . . . . . 9 (((𝑤𝑢) ∈ 𝑈 ∧ (𝑎𝑏) = ((𝑤𝑢) “ {𝑝})) → ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝}))
176, 13, 16syl2anc 692 . . . . . . . 8 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝}))
18 vex 3194 . . . . . . . . . . 11 𝑎 ∈ V
1918inex1 4764 . . . . . . . . . 10 (𝑎𝑏) ∈ V
20 utopustuq.1 . . . . . . . . . . 11 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
2120ustuqtoplem 21948 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ (𝑎𝑏) ∈ V) → ((𝑎𝑏) ∈ (𝑁𝑝) ↔ ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝})))
2219, 21mpan2 706 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ((𝑎𝑏) ∈ (𝑁𝑝) ↔ ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝})))
2322biimpar 502 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ ∃𝑥𝑈 (𝑎𝑏) = (𝑥 “ {𝑝})) → (𝑎𝑏) ∈ (𝑁𝑝))
241, 17, 23syl2anc 692 . . . . . . 7 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎𝑏) ∈ (𝑁𝑝))
25 simp-4l 805 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋))
26 simpllr 798 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑏 ∈ (𝑁𝑝))
27 vex 3194 . . . . . . . . . 10 𝑏 ∈ V
2820ustuqtoplem 21948 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏 ∈ V) → (𝑏 ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝})))
2927, 28mpan2 706 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑏 ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝})))
3029biimpa 501 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏 ∈ (𝑁𝑝)) → ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝}))
3125, 26, 30syl2anc 692 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝}))
3224, 31r19.29a 3076 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑎𝑏) ∈ (𝑁𝑝))
3320ustuqtoplem 21948 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝})))
3418, 33mpan2 706 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝})))
3534biimpa 501 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝}))
3635adantr 481 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) → ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝}))
3732, 36r19.29a 3076 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑏 ∈ (𝑁𝑝)) → (𝑎𝑏) ∈ (𝑁𝑝))
3837ralrimiva 2965 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∀𝑏 ∈ (𝑁𝑝)(𝑎𝑏) ∈ (𝑁𝑝))
3938ralrimiva 2965 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → ∀𝑎 ∈ (𝑁𝑝)∀𝑏 ∈ (𝑁𝑝)(𝑎𝑏) ∈ (𝑁𝑝))
40 fvex 6160 . . . 4 (𝑁𝑝) ∈ V
41 inficl 8276 . . . 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 3641 . 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 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  Vcvv 3191  cin 3559  wss 3560  {csn 4153  cmpt 4678  ran crn 5080  cima 5082  cfv 5850  ficfi 8261  UnifOncust 21908
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-fin 7904  df-fi 8262  df-ust 21909
This theorem is referenced by:  ustuqtop  21955  utopsnneiplem  21956
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