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Theorem ustuqtop1 21855
 Description: Lemma for ustuqtop 21860, similar to ssnei2 20730. (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 1105 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ (𝑎 ∈ (𝑁𝑝) ∧ 𝑤𝑈𝑎 = (𝑤 “ {𝑝}))) → 𝑈 ∈ (UnifOn‘𝑋))
213anassrs 1282 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑈 ∈ (UnifOn‘𝑋))
3 simplr 788 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑤𝑈)
4 ustssxp 21818 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
52, 3, 4syl2anc 691 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑤 ⊆ (𝑋 × 𝑋))
6 simpl1r 1106 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ (𝑎 ∈ (𝑁𝑝) ∧ 𝑤𝑈𝑎 = (𝑤 “ {𝑝}))) → 𝑝𝑋)
763anassrs 1282 . . . . . . . 8 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑝𝑋)
87snssd 4281 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → {𝑝} ⊆ 𝑋)
9 simpl3 1059 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ (𝑎 ∈ (𝑁𝑝) ∧ 𝑤𝑈𝑎 = (𝑤 “ {𝑝}))) → 𝑏𝑋)
1093anassrs 1282 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑏𝑋)
11 xpss12 5148 . . . . . . 7 (({𝑝} ⊆ 𝑋𝑏𝑋) → ({𝑝} × 𝑏) ⊆ (𝑋 × 𝑋))
128, 10, 11syl2anc 691 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → ({𝑝} × 𝑏) ⊆ (𝑋 × 𝑋))
135, 12unssd 3751 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑤 ∪ ({𝑝} × 𝑏)) ⊆ (𝑋 × 𝑋))
14 ssun1 3738 . . . . . 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 21819 . . . . . 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 1321 . . . 4 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑤 ∪ ({𝑝} × 𝑏)) ∈ 𝑈)
19 simpl2 1058 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ (𝑎 ∈ (𝑁𝑝) ∧ 𝑤𝑈𝑎 = (𝑤 “ {𝑝}))) → 𝑎𝑏)
20193anassrs 1282 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑎𝑏)
21 ssequn1 3745 . . . . . . 7 (𝑎𝑏 ↔ (𝑎𝑏) = 𝑏)
2221biimpi 205 . . . . . 6 (𝑎𝑏 → (𝑎𝑏) = 𝑏)
23 id 22 . . . . . . . 8 (𝑎 = (𝑤 “ {𝑝}) → 𝑎 = (𝑤 “ {𝑝}))
24 inidm 3784 . . . . . . . . . . 11 ({𝑝} ∩ {𝑝}) = {𝑝}
25 vex 3176 . . . . . . . . . . . 12 𝑝 ∈ V
2625snnz 4252 . . . . . . . . . . 11 {𝑝} ≠ ∅
2724, 26eqnetri 2852 . . . . . . . . . 10 ({𝑝} ∩ {𝑝}) ≠ ∅
28 xpima2 5497 . . . . . . . . . 10 (({𝑝} ∩ {𝑝}) ≠ ∅ → (({𝑝} × 𝑏) “ {𝑝}) = 𝑏)
2927, 28mp1i 13 . . . . . . . . 9 (𝑎 = (𝑤 “ {𝑝}) → (({𝑝} × 𝑏) “ {𝑝}) = 𝑏)
3029eqcomd 2616 . . . . . . . 8 (𝑎 = (𝑤 “ {𝑝}) → 𝑏 = (({𝑝} × 𝑏) “ {𝑝}))
3123, 30uneq12d 3730 . . . . . . 7 (𝑎 = (𝑤 “ {𝑝}) → (𝑎𝑏) = ((𝑤 “ {𝑝}) ∪ (({𝑝} × 𝑏) “ {𝑝})))
32 imaundir 5465 . . . . . . 7 ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}) = ((𝑤 “ {𝑝}) ∪ (({𝑝} × 𝑏) “ {𝑝}))
3331, 32syl6eqr 2662 . . . . . 6 (𝑎 = (𝑤 “ {𝑝}) → (𝑎𝑏) = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
3422, 33sylan9req 2665 . . . . 5 ((𝑎𝑏𝑎 = (𝑤 “ {𝑝})) → 𝑏 = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
3520, 34sylancom 698 . . . 4 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑏 = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
36 imaeq1 5380 . . . . . 6 (𝑢 = (𝑤 ∪ ({𝑝} × 𝑏)) → (𝑢 “ {𝑝}) = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝}))
3736eqeq2d 2620 . . . . 5 (𝑢 = (𝑤 ∪ ({𝑝} × 𝑏)) → (𝑏 = (𝑢 “ {𝑝}) ↔ 𝑏 = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝})))
3837rspcev 3282 . . . 4 (((𝑤 ∪ ({𝑝} × 𝑏)) ∈ 𝑈𝑏 = ((𝑤 ∪ ({𝑝} × 𝑏)) “ {𝑝})) → ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝}))
3918, 35, 38syl2anc 691 . . 3 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) ∧ 𝑤𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝}))
40 vex 3176 . . . . . 6 𝑎 ∈ V
41 utopustuq.1 . . . . . . 7 𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))
4241ustuqtoplem 21853 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁𝑝) ↔ ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝})))
4340, 42mpan2 703 . . . . 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 1216 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑤𝑈 𝑎 = (𝑤 “ {𝑝}))
4639, 45r19.29a 3060 . 2 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝}))
47 vex 3176 . . . . 5 𝑏 ∈ V
4841ustuqtoplem 21853 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑏 ∈ V) → (𝑏 ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝})))
4947, 48mpan2 703 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (𝑏 ∈ (𝑁𝑝) ↔ ∃𝑢𝑈 𝑏 = (𝑢 “ {𝑝})))
50493ad2ant1 1075 . . 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 246 1 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125   ↦ cmpt 4643   × cxp 5036  ran crn 5039   “ cima 5041  ‘cfv 5804  UnifOncust 21813 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ust 21814 This theorem is referenced by:  ustuqtop4  21858  ustuqtop  21860  utopsnneiplem  21861
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