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Theorem dfac14lem 23625
Description: Lemma for dfac14 23626. By equipping 𝑆 ∪ {𝑃} for some 𝑃𝑆 with the particular point topology, we can show that 𝑃 is in the closure of 𝑆; hence the sequence 𝑃(𝑥) is in the product of the closures, and we can utilize this instance of ptcls 23624 to extract an element of the closure of X𝑘𝐼𝑆. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypotheses
Ref Expression
dfac14lem.i (𝜑𝐼𝑉)
dfac14lem.s ((𝜑𝑥𝐼) → 𝑆𝑊)
dfac14lem.0 ((𝜑𝑥𝐼) → 𝑆 ≠ ∅)
dfac14lem.p 𝑃 = 𝒫 𝑆
dfac14lem.r 𝑅 = {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))}
dfac14lem.j 𝐽 = (∏t‘(𝑥𝐼𝑅))
dfac14lem.c (𝜑 → ((cls‘𝐽)‘X𝑥𝐼 𝑆) = X𝑥𝐼 ((cls‘𝑅)‘𝑆))
Assertion
Ref Expression
dfac14lem (𝜑X𝑥𝐼 𝑆 ≠ ∅)
Distinct variable groups:   𝑥,𝐼   𝑦,𝑃   𝜑,𝑥   𝑦,𝑆
Allowed substitution hints:   𝜑(𝑦)   𝑃(𝑥)   𝑅(𝑥,𝑦)   𝑆(𝑥)   𝐼(𝑦)   𝐽(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem dfac14lem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eleq2w 2825 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑃𝑦𝑃𝑧))
2 eqeq1 2741 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦 = (𝑆 ∪ {𝑃}) ↔ 𝑧 = (𝑆 ∪ {𝑃})))
31, 2imbi12d 344 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝑃𝑦𝑦 = (𝑆 ∪ {𝑃})) ↔ (𝑃𝑧𝑧 = (𝑆 ∪ {𝑃}))))
4 dfac14lem.r . . . . . . . . . 10 𝑅 = {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))}
53, 4elrab2 3695 . . . . . . . . 9 (𝑧𝑅 ↔ (𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∧ (𝑃𝑧𝑧 = (𝑆 ∪ {𝑃}))))
6 dfac14lem.0 . . . . . . . . . . . . 13 ((𝜑𝑥𝐼) → 𝑆 ≠ ∅)
76adantr 480 . . . . . . . . . . . 12 (((𝜑𝑥𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → 𝑆 ≠ ∅)
8 ineq1 4213 . . . . . . . . . . . . . 14 (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧𝑆) = ((𝑆 ∪ {𝑃}) ∩ 𝑆))
9 ssun1 4178 . . . . . . . . . . . . . . 15 𝑆 ⊆ (𝑆 ∪ {𝑃})
10 sseqin2 4223 . . . . . . . . . . . . . . 15 (𝑆 ⊆ (𝑆 ∪ {𝑃}) ↔ ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆)
119, 10mpbi 230 . . . . . . . . . . . . . 14 ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆
128, 11eqtrdi 2793 . . . . . . . . . . . . 13 (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧𝑆) = 𝑆)
1312neeq1d 3000 . . . . . . . . . . . 12 (𝑧 = (𝑆 ∪ {𝑃}) → ((𝑧𝑆) ≠ ∅ ↔ 𝑆 ≠ ∅))
147, 13syl5ibrcom 247 . . . . . . . . . . 11 (((𝜑𝑥𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧𝑆) ≠ ∅))
1514imim2d 57 . . . . . . . . . 10 (((𝜑𝑥𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → ((𝑃𝑧𝑧 = (𝑆 ∪ {𝑃})) → (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
1615expimpd 453 . . . . . . . . 9 ((𝜑𝑥𝐼) → ((𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∧ (𝑃𝑧𝑧 = (𝑆 ∪ {𝑃}))) → (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
175, 16biimtrid 242 . . . . . . . 8 ((𝜑𝑥𝐼) → (𝑧𝑅 → (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
1817ralrimiv 3145 . . . . . . 7 ((𝜑𝑥𝐼) → ∀𝑧𝑅 (𝑃𝑧 → (𝑧𝑆) ≠ ∅))
19 dfac14lem.s . . . . . . . . . . . 12 ((𝜑𝑥𝐼) → 𝑆𝑊)
20 snex 5436 . . . . . . . . . . . 12 {𝑃} ∈ V
21 unexg 7763 . . . . . . . . . . . 12 ((𝑆𝑊 ∧ {𝑃} ∈ V) → (𝑆 ∪ {𝑃}) ∈ V)
2219, 20, 21sylancl 586 . . . . . . . . . . 11 ((𝜑𝑥𝐼) → (𝑆 ∪ {𝑃}) ∈ V)
23 ssun2 4179 . . . . . . . . . . . 12 {𝑃} ⊆ (𝑆 ∪ {𝑃})
24 dfac14lem.p . . . . . . . . . . . . . 14 𝑃 = 𝒫 𝑆
25 uniexg 7760 . . . . . . . . . . . . . . 15 (𝑆𝑊 𝑆 ∈ V)
26 pwexg 5378 . . . . . . . . . . . . . . 15 ( 𝑆 ∈ V → 𝒫 𝑆 ∈ V)
2719, 25, 263syl 18 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐼) → 𝒫 𝑆 ∈ V)
2824, 27eqeltrid 2845 . . . . . . . . . . . . 13 ((𝜑𝑥𝐼) → 𝑃 ∈ V)
29 snidg 4660 . . . . . . . . . . . . 13 (𝑃 ∈ V → 𝑃 ∈ {𝑃})
3028, 29syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝐼) → 𝑃 ∈ {𝑃})
3123, 30sselid 3981 . . . . . . . . . . 11 ((𝜑𝑥𝐼) → 𝑃 ∈ (𝑆 ∪ {𝑃}))
32 epttop 23016 . . . . . . . . . . 11 (((𝑆 ∪ {𝑃}) ∈ V ∧ 𝑃 ∈ (𝑆 ∪ {𝑃})) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
3322, 31, 32syl2anc 584 . . . . . . . . . 10 ((𝜑𝑥𝐼) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
344, 33eqeltrid 2845 . . . . . . . . 9 ((𝜑𝑥𝐼) → 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
35 topontop 22919 . . . . . . . . 9 (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → 𝑅 ∈ Top)
3634, 35syl 17 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑅 ∈ Top)
37 toponuni 22920 . . . . . . . . . 10 (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → (𝑆 ∪ {𝑃}) = 𝑅)
3834, 37syl 17 . . . . . . . . 9 ((𝜑𝑥𝐼) → (𝑆 ∪ {𝑃}) = 𝑅)
399, 38sseqtrid 4026 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑆 𝑅)
4031, 38eleqtrd 2843 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑃 𝑅)
41 eqid 2737 . . . . . . . . 9 𝑅 = 𝑅
4241elcls 23081 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 𝑅𝑃 𝑅) → (𝑃 ∈ ((cls‘𝑅)‘𝑆) ↔ ∀𝑧𝑅 (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
4336, 39, 40, 42syl3anc 1373 . . . . . . 7 ((𝜑𝑥𝐼) → (𝑃 ∈ ((cls‘𝑅)‘𝑆) ↔ ∀𝑧𝑅 (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
4418, 43mpbird 257 . . . . . 6 ((𝜑𝑥𝐼) → 𝑃 ∈ ((cls‘𝑅)‘𝑆))
4544ralrimiva 3146 . . . . 5 (𝜑 → ∀𝑥𝐼 𝑃 ∈ ((cls‘𝑅)‘𝑆))
46 dfac14lem.i . . . . . 6 (𝜑𝐼𝑉)
47 mptelixpg 8975 . . . . . 6 (𝐼𝑉 → ((𝑥𝐼𝑃) ∈ X𝑥𝐼 ((cls‘𝑅)‘𝑆) ↔ ∀𝑥𝐼 𝑃 ∈ ((cls‘𝑅)‘𝑆)))
4846, 47syl 17 . . . . 5 (𝜑 → ((𝑥𝐼𝑃) ∈ X𝑥𝐼 ((cls‘𝑅)‘𝑆) ↔ ∀𝑥𝐼 𝑃 ∈ ((cls‘𝑅)‘𝑆)))
4945, 48mpbird 257 . . . 4 (𝜑 → (𝑥𝐼𝑃) ∈ X𝑥𝐼 ((cls‘𝑅)‘𝑆))
5049ne0d 4342 . . 3 (𝜑X𝑥𝐼 ((cls‘𝑅)‘𝑆) ≠ ∅)
51 dfac14lem.c . . 3 (𝜑 → ((cls‘𝐽)‘X𝑥𝐼 𝑆) = X𝑥𝐼 ((cls‘𝑅)‘𝑆))
5234ralrimiva 3146 . . . . 5 (𝜑 → ∀𝑥𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
53 dfac14lem.j . . . . . 6 𝐽 = (∏t‘(𝑥𝐼𝑅))
5453pttopon 23604 . . . . 5 ((𝐼𝑉 ∧ ∀𝑥𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃}))) → 𝐽 ∈ (TopOn‘X𝑥𝐼 (𝑆 ∪ {𝑃})))
5546, 52, 54syl2anc 584 . . . 4 (𝜑𝐽 ∈ (TopOn‘X𝑥𝐼 (𝑆 ∪ {𝑃})))
56 topontop 22919 . . . 4 (𝐽 ∈ (TopOn‘X𝑥𝐼 (𝑆 ∪ {𝑃})) → 𝐽 ∈ Top)
57 cls0 23088 . . . 4 (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
5855, 56, 573syl 18 . . 3 (𝜑 → ((cls‘𝐽)‘∅) = ∅)
5950, 51, 583netr4d 3018 . 2 (𝜑 → ((cls‘𝐽)‘X𝑥𝐼 𝑆) ≠ ((cls‘𝐽)‘∅))
60 fveq2 6906 . . 3 (X𝑥𝐼 𝑆 = ∅ → ((cls‘𝐽)‘X𝑥𝐼 𝑆) = ((cls‘𝐽)‘∅))
6160necon3i 2973 . 2 (((cls‘𝐽)‘X𝑥𝐼 𝑆) ≠ ((cls‘𝐽)‘∅) → X𝑥𝐼 𝑆 ≠ ∅)
6259, 61syl 17 1 (𝜑X𝑥𝐼 𝑆 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  {crab 3436  Vcvv 3480  cun 3949  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   cuni 4907  cmpt 5225  cfv 6561  Xcixp 8937  tcpt 17483  Topctop 22899  TopOnctopon 22916  clsccl 23026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-2o 8507  df-ixp 8938  df-en 8986  df-fin 8989  df-fi 9451  df-topgen 17488  df-pt 17489  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029
This theorem is referenced by:  dfac14  23626
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