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Theorem ustuqtop1 24067
Description: Lemma for ustuqtop 24072, similar to ssnei2 22941. (Contributed by Thierry Arnoux, 11-Jan-2018.)
Hypothesis
Ref Expression
utopustuq.1 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})))
Assertion
Ref Expression
ustuqtop1 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ 𝑏 ∈ (π‘β€˜π‘))
Distinct variable groups:   𝑣,𝑝,π‘ˆ   𝑋,𝑝,𝑣   π‘Ž,𝑏,𝑝,𝑁   𝑣,π‘Ž,π‘ˆ,𝑏   𝑋,π‘Ž,𝑏
Allowed substitution hint:   𝑁(𝑣)
Proof of Theorem ustuqtop1
Dummy variables 𝑀 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1l 1221 . . . . . 6 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ (π‘Ž ∈ (π‘β€˜π‘) ∧ 𝑀 ∈ π‘ˆ ∧ π‘Ž = (𝑀 β€œ {𝑝}))) β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
213anassrs 1357 . . . . 5 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
3 simplr 766 . . . . 5 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) β†’ 𝑀 ∈ π‘ˆ)
4 ustssxp 24030 . . . . . . 7 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑀 ∈ π‘ˆ) β†’ 𝑀 βŠ† (𝑋 Γ— 𝑋))
52, 3, 4syl2anc 583 . . . . . 6 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) β†’ 𝑀 βŠ† (𝑋 Γ— 𝑋))
6 simpl1r 1222 . . . . . . . . 9 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ (π‘Ž ∈ (π‘β€˜π‘) ∧ 𝑀 ∈ π‘ˆ ∧ π‘Ž = (𝑀 β€œ {𝑝}))) β†’ 𝑝 ∈ 𝑋)
763anassrs 1357 . . . . . . . 8 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) β†’ 𝑝 ∈ 𝑋)
87snssd 4804 . . . . . . 7 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) β†’ {𝑝} βŠ† 𝑋)
9 simpl3 1190 . . . . . . . 8 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ (π‘Ž ∈ (π‘β€˜π‘) ∧ 𝑀 ∈ π‘ˆ ∧ π‘Ž = (𝑀 β€œ {𝑝}))) β†’ 𝑏 βŠ† 𝑋)
1093anassrs 1357 . . . . . . 7 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) β†’ 𝑏 βŠ† 𝑋)
11 xpss12 5681 . . . . . . 7 (({𝑝} βŠ† 𝑋 ∧ 𝑏 βŠ† 𝑋) β†’ ({𝑝} Γ— 𝑏) βŠ† (𝑋 Γ— 𝑋))
128, 10, 11syl2anc 583 . . . . . 6 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) β†’ ({𝑝} Γ— 𝑏) βŠ† (𝑋 Γ— 𝑋))
135, 12unssd 4178 . . . . 5 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) β†’ (𝑀 βˆͺ ({𝑝} Γ— 𝑏)) βŠ† (𝑋 Γ— 𝑋))
14 ssun1 4164 . . . . . 6 𝑀 βŠ† (𝑀 βˆͺ ({𝑝} Γ— 𝑏))
1514a1i 11 . . . . 5 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) β†’ 𝑀 βŠ† (𝑀 βˆͺ ({𝑝} Γ— 𝑏)))
16 ustssel 24031 . . . . . 6 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑀 ∈ π‘ˆ ∧ (𝑀 βˆͺ ({𝑝} Γ— 𝑏)) βŠ† (𝑋 Γ— 𝑋)) β†’ (𝑀 βŠ† (𝑀 βˆͺ ({𝑝} Γ— 𝑏)) β†’ (𝑀 βˆͺ ({𝑝} Γ— 𝑏)) ∈ π‘ˆ))
1716imp 406 . . . . 5 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑀 ∈ π‘ˆ ∧ (𝑀 βˆͺ ({𝑝} Γ— 𝑏)) βŠ† (𝑋 Γ— 𝑋)) ∧ 𝑀 βŠ† (𝑀 βˆͺ ({𝑝} Γ— 𝑏))) β†’ (𝑀 βˆͺ ({𝑝} Γ— 𝑏)) ∈ π‘ˆ)
182, 3, 13, 15, 17syl31anc 1370 . . . 4 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) β†’ (𝑀 βˆͺ ({𝑝} Γ— 𝑏)) ∈ π‘ˆ)
19 simpl2 1189 . . . . . 6 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ (π‘Ž ∈ (π‘β€˜π‘) ∧ 𝑀 ∈ π‘ˆ ∧ π‘Ž = (𝑀 β€œ {𝑝}))) β†’ π‘Ž βŠ† 𝑏)
20193anassrs 1357 . . . . 5 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) β†’ π‘Ž βŠ† 𝑏)
21 ssequn1 4172 . . . . . . 7 (π‘Ž βŠ† 𝑏 ↔ (π‘Ž βˆͺ 𝑏) = 𝑏)
2221biimpi 215 . . . . . 6 (π‘Ž βŠ† 𝑏 β†’ (π‘Ž βˆͺ 𝑏) = 𝑏)
23 id 22 . . . . . . . 8 (π‘Ž = (𝑀 β€œ {𝑝}) β†’ π‘Ž = (𝑀 β€œ {𝑝}))
24 inidm 4210 . . . . . . . . . . 11 ({𝑝} ∩ {𝑝}) = {𝑝}
25 vex 3470 . . . . . . . . . . . 12 𝑝 ∈ V
2625snnz 4772 . . . . . . . . . . 11 {𝑝} β‰  βˆ…
2724, 26eqnetri 3003 . . . . . . . . . 10 ({𝑝} ∩ {𝑝}) β‰  βˆ…
28 xpima2 6173 . . . . . . . . . 10 (({𝑝} ∩ {𝑝}) β‰  βˆ… β†’ (({𝑝} Γ— 𝑏) β€œ {𝑝}) = 𝑏)
2927, 28mp1i 13 . . . . . . . . 9 (π‘Ž = (𝑀 β€œ {𝑝}) β†’ (({𝑝} Γ— 𝑏) β€œ {𝑝}) = 𝑏)
3029eqcomd 2730 . . . . . . . 8 (π‘Ž = (𝑀 β€œ {𝑝}) β†’ 𝑏 = (({𝑝} Γ— 𝑏) β€œ {𝑝}))
3123, 30uneq12d 4156 . . . . . . 7 (π‘Ž = (𝑀 β€œ {𝑝}) β†’ (π‘Ž βˆͺ 𝑏) = ((𝑀 β€œ {𝑝}) βˆͺ (({𝑝} Γ— 𝑏) β€œ {𝑝})))
32 imaundir 6140 . . . . . . 7 ((𝑀 βˆͺ ({𝑝} Γ— 𝑏)) β€œ {𝑝}) = ((𝑀 β€œ {𝑝}) βˆͺ (({𝑝} Γ— 𝑏) β€œ {𝑝}))
3331, 32eqtr4di 2782 . . . . . 6 (π‘Ž = (𝑀 β€œ {𝑝}) β†’ (π‘Ž βˆͺ 𝑏) = ((𝑀 βˆͺ ({𝑝} Γ— 𝑏)) β€œ {𝑝}))
3422, 33sylan9req 2785 . . . . 5 ((π‘Ž βŠ† 𝑏 ∧ π‘Ž = (𝑀 β€œ {𝑝})) β†’ 𝑏 = ((𝑀 βˆͺ ({𝑝} Γ— 𝑏)) β€œ {𝑝}))
3520, 34sylancom 587 . . . 4 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) β†’ 𝑏 = ((𝑀 βˆͺ ({𝑝} Γ— 𝑏)) β€œ {𝑝}))
36 imaeq1 6044 . . . . 5 (𝑒 = (𝑀 βˆͺ ({𝑝} Γ— 𝑏)) β†’ (𝑒 β€œ {𝑝}) = ((𝑀 βˆͺ ({𝑝} Γ— 𝑏)) β€œ {𝑝}))
3736rspceeqv 3625 . . . 4 (((𝑀 βˆͺ ({𝑝} Γ— 𝑏)) ∈ π‘ˆ ∧ 𝑏 = ((𝑀 βˆͺ ({𝑝} Γ— 𝑏)) β€œ {𝑝})) β†’ βˆƒπ‘’ ∈ π‘ˆ 𝑏 = (𝑒 β€œ {𝑝}))
3818, 35, 37syl2anc 583 . . 3 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) ∧ 𝑀 ∈ π‘ˆ) ∧ π‘Ž = (𝑀 β€œ {𝑝})) β†’ βˆƒπ‘’ ∈ π‘ˆ 𝑏 = (𝑒 β€œ {𝑝}))
39 utopustuq.1 . . . . . . 7 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ π‘ˆ ↦ (𝑣 β€œ {𝑝})))
4039ustuqtoplem 24065 . . . . . 6 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ V) β†’ (π‘Ž ∈ (π‘β€˜π‘) ↔ βˆƒπ‘€ ∈ π‘ˆ π‘Ž = (𝑀 β€œ {𝑝})))
4140elvd 3473 . . . . 5 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ (π‘Ž ∈ (π‘β€˜π‘) ↔ βˆƒπ‘€ ∈ π‘ˆ π‘Ž = (𝑀 β€œ {𝑝})))
4241biimpa 476 . . . 4 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ βˆƒπ‘€ ∈ π‘ˆ π‘Ž = (𝑀 β€œ {𝑝}))
43423ad2antl1 1182 . . 3 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ βˆƒπ‘€ ∈ π‘ˆ π‘Ž = (𝑀 β€œ {𝑝}))
4438, 43r19.29a 3154 . 2 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ βˆƒπ‘’ ∈ π‘ˆ 𝑏 = (𝑒 β€œ {𝑝}))
4539ustuqtoplem 24065 . . . . 5 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 ∈ V) β†’ (𝑏 ∈ (π‘β€˜π‘) ↔ βˆƒπ‘’ ∈ π‘ˆ 𝑏 = (𝑒 β€œ {𝑝})))
4645elvd 3473 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) β†’ (𝑏 ∈ (π‘β€˜π‘) ↔ βˆƒπ‘’ ∈ π‘ˆ 𝑏 = (𝑒 β€œ {𝑝})))
47463ad2ant1 1130 . . 3 (((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) β†’ (𝑏 ∈ (π‘β€˜π‘) ↔ βˆƒπ‘’ ∈ π‘ˆ 𝑏 = (𝑒 β€œ {𝑝})))
4847adantr 480 . 2 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ (𝑏 ∈ (π‘β€˜π‘) ↔ βˆƒπ‘’ ∈ π‘ˆ 𝑏 = (𝑒 β€œ {𝑝})))
4944, 48mpbird 257 1 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑝 ∈ 𝑋) ∧ π‘Ž βŠ† 𝑏 ∧ 𝑏 βŠ† 𝑋) ∧ π‘Ž ∈ (π‘β€˜π‘)) β†’ 𝑏 ∈ (π‘β€˜π‘))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2932  βˆƒwrex 3062  Vcvv 3466   βˆͺ cun 3938   ∩ cin 3939   βŠ† wss 3940  βˆ…c0 4314  {csn 4620   ↦ cmpt 5221   Γ— cxp 5664  ran crn 5667   β€œ cima 5669  β€˜cfv 6533  UnifOncust 24025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ust 24026
This theorem is referenced by:  ustuqtop4  24070  ustuqtop  24072  utopsnneiplem  24073
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