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Theorem ustuqtop1 22092
Description: Lemma for ustuqtop 22097, similar to ssnei2 20968. (Contributed by Thierry Arnoux, 11-Jan-2018.)
Hypothesis
Ref Expression
utopustuq.1 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
Assertion
Ref Expression
ustuqtop1 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))
Distinct variable groups:   𝑣,𝑝,𝑈   𝑋,𝑝,𝑣   𝑎,𝑏,𝑝,𝑁   𝑣,𝑎,𝑈,𝑏   𝑋,𝑎,𝑏
Allowed substitution hint:   𝑁(𝑣)

Proof of Theorem ustuqtop1
Dummy variables 𝑤 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1l 1132 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ (𝑎 ∈ (𝑁𝑝) ∧ 𝑤𝑈𝑎 = (𝑤 “ {𝑝}))) → 𝑈 ∈ (UnifOn‘𝑋))
213anassrs 1313 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑈 ∈ (UnifOn‘𝑋))
3 simplr 807 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑤𝑈)
4 ustssxp 22055 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
52, 3, 4syl2anc 694 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑤 ⊆ (𝑋 × 𝑋))
6 simpl1r 1133 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ (𝑎 ∈ (𝑁𝑝) ∧ 𝑤𝑈𝑎 = (𝑤 “ {𝑝}))) → 𝑝𝑋)
763anassrs 1313 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑝𝑋)
87snssd 4372 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → {𝑝} ⊆ 𝑋)
9 simpl3 1086 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ (𝑎 ∈ (𝑁𝑝) ∧ 𝑤𝑈𝑎 = (𝑤 “ {𝑝}))) → 𝑏𝑋)
1093anassrs 1313 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑏𝑋)
11 xpss12 5158 . . . . . . 7 (({𝑝} ⊆ 𝑋𝑏𝑋) → ({𝑝} × 𝑏) ⊆ (𝑋 × 𝑋))
128, 10, 11syl2anc 694 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → ({𝑝} × 𝑏) ⊆ (𝑋 × 𝑋))
135, 12unssd 3822 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑤 ∪ ({𝑝} × 𝑏)) ⊆ (𝑋 × 𝑋))
14 ssun1 3809 . . . . . 6 𝑤 ⊆ (𝑤 ∪ ({𝑝} × 𝑏))
1514a1i 11 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑤 ⊆ (𝑤 ∪ ({𝑝} × 𝑏)))
16 ustssel 22056 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈 ∧ (𝑤 ∪ ({𝑝} × 𝑏)) ⊆ (𝑋 × 𝑋)) → (𝑤 ⊆ (𝑤 ∪ ({𝑝} × 𝑏)) → (𝑤 ∪ ({𝑝} × 𝑏)) ∈ 𝑈))
1716imp 444 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈 ∧ (𝑤 ∪ ({𝑝} × 𝑏)) ⊆ (𝑋 × 𝑋)) ∧ 𝑤 ⊆ (𝑤 ∪ ({𝑝} × 𝑏))) → (𝑤 ∪ ({𝑝} × 𝑏)) ∈ 𝑈)
182, 3, 13, 15, 17syl31anc 1369 . . . 4 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑤 ∪ ({𝑝} × 𝑏)) ∈ 𝑈)
19 simpl2 1085 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ (𝑎 ∈ (𝑁𝑝) ∧ 𝑤𝑈𝑎 = (𝑤 “ {𝑝}))) → 𝑎𝑏)
20193anassrs 1313 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑎𝑏)
21 ssequn1 3816 . . . . . . 7 (𝑎𝑏 ↔ (𝑎𝑏) = 𝑏)
2221biimpi 206 . . . . . 6 (𝑎𝑏 → (𝑎𝑏) = 𝑏)
23 id 22 . . . . . . . 8 (𝑎 = (𝑤 “ {𝑝}) → 𝑎 = (𝑤 “ {𝑝}))
24 inidm 3855 . . . . . . . . . . 11 ({𝑝} ∩ {𝑝}) = {𝑝}
25 vex 3234 . . . . . . . . . . . 12 𝑝 ∈ V
2625snnz 4340 . . . . . . . . . . 11 {𝑝} ≠ ∅
2724, 26eqnetri 2893 . . . . . . . . . 10 ({𝑝} ∩ {𝑝}) ≠ ∅
28 xpima2 5613 . . . . . . . . . 10 (({𝑝} ∩ {𝑝}) ≠ ∅ → (({𝑝} × 𝑏) “ {𝑝}) = 𝑏)
2927, 28mp1i 13 . . . . . . . . 9 (𝑎 = (𝑤 “ {𝑝}) → (({𝑝} × 𝑏) “ {𝑝}) = 𝑏)
3029eqcomd 2657 . . . . . . . 8 (𝑎 = (𝑤 “ {𝑝}) → 𝑏 = (({𝑝} × 𝑏) “ {𝑝}))
3123, 30uneq12d 3801 . . . . . . 7 (𝑎 = (𝑤 “ {𝑝}) → (𝑎𝑏) = ((𝑤 “ {𝑝}) ∪ (({𝑝} × 𝑏) “ {𝑝})))
32 imaundir 5581 . . . . . . 7 ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}) = ((𝑤 “ {𝑝}) ∪ (({𝑝} × 𝑏) “ {𝑝}))
3331, 32syl6eqr 2703 . . . . . 6 (𝑎 = (𝑤 “ {𝑝}) → (𝑎𝑏) = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
3422, 33sylan9req 2706 . . . . 5 ((𝑎𝑏𝑎 = (𝑤 “ {𝑝})) → 𝑏 = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
3520, 34sylancom 702 . . . 4 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑏 = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
36 imaeq1 5496 . . . . . 6 (𝑢 = (𝑤 ∪ ({𝑝} × 𝑏)) → (𝑢 “ {𝑝}) = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
3736eqeq2d 2661 . . . . 5 (𝑢 = (𝑤 ∪ ({𝑝} × 𝑏)) → (𝑏 = (𝑢 “ {𝑝}) ↔ 𝑏 = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝})))
3837rspcev 3340 . . . 4 (((𝑤 ∪ ({𝑝} × 𝑏)) ∈ 𝑈𝑏 = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝})) → ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝}))
3918, 35, 38syl2anc 694 . . 3 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝}))
40 vex 3234 . . . . . 6 𝑎 ∈ V
41 utopustuq.1 . . . . . . 7 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
4241ustuqtoplem 22090 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝})))
4340, 42mpan2 707 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝})))
4443biimpa 500 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝}))
45443ad2antl1 1243 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝}))
4639, 45r19.29a 3107 . 2 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝}))
47 vex 3234 . . . . 5 𝑏 ∈ V
4841ustuqtoplem 22090 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏 ∈ V) → (𝑏 ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝})))
4947, 48mpan2 707 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑏 ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝})))
50493ad2ant1 1102 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) → (𝑏 ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝})))
5150adantr 480 . 2 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → (𝑏 ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝})))
5246, 51mpbird 247 1 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wrex 2942  Vcvv 3231  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210  cmpt 4762   × cxp 5141  ran crn 5144  cima 5146  cfv 5926  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-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-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  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-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ust 22051
This theorem is referenced by:  ustuqtop4  22095  ustuqtop  22097  utopsnneiplem  22098
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