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Theorem dfac14lem 23607
Description: Lemma for dfac14 23608. 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 23606 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 2810 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑃𝑦𝑃𝑧))
2 eqeq1 2730 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦 = (𝑆 ∪ {𝑃}) ↔ 𝑧 = (𝑆 ∪ {𝑃})))
31, 2imbi12d 343 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝑃𝑦𝑦 = (𝑆 ∪ {𝑃})) ↔ (𝑃𝑧𝑧 = (𝑆 ∪ {𝑃}))))
4 dfac14lem.r . . . . . . . . . 10 𝑅 = {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))}
53, 4elrab2 3684 . . . . . . . . 9 (𝑧𝑅 ↔ (𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∧ (𝑃𝑧𝑧 = (𝑆 ∪ {𝑃}))))
6 dfac14lem.0 . . . . . . . . . . . . 13 ((𝜑𝑥𝐼) → 𝑆 ≠ ∅)
76adantr 479 . . . . . . . . . . . 12 (((𝜑𝑥𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → 𝑆 ≠ ∅)
8 ineq1 4204 . . . . . . . . . . . . . 14 (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧𝑆) = ((𝑆 ∪ {𝑃}) ∩ 𝑆))
9 ssun1 4171 . . . . . . . . . . . . . . 15 𝑆 ⊆ (𝑆 ∪ {𝑃})
10 sseqin2 4214 . . . . . . . . . . . . . . 15 (𝑆 ⊆ (𝑆 ∪ {𝑃}) ↔ ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆)
119, 10mpbi 229 . . . . . . . . . . . . . 14 ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆
128, 11eqtrdi 2782 . . . . . . . . . . . . 13 (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧𝑆) = 𝑆)
1312neeq1d 2990 . . . . . . . . . . . 12 (𝑧 = (𝑆 ∪ {𝑃}) → ((𝑧𝑆) ≠ ∅ ↔ 𝑆 ≠ ∅))
147, 13syl5ibrcom 246 . . . . . . . . . . 11 (((𝜑𝑥𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧𝑆) ≠ ∅))
1514imim2d 57 . . . . . . . . . 10 (((𝜑𝑥𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → ((𝑃𝑧𝑧 = (𝑆 ∪ {𝑃})) → (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
1615expimpd 452 . . . . . . . . 9 ((𝜑𝑥𝐼) → ((𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∧ (𝑃𝑧𝑧 = (𝑆 ∪ {𝑃}))) → (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
175, 16biimtrid 241 . . . . . . . 8 ((𝜑𝑥𝐼) → (𝑧𝑅 → (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
1817ralrimiv 3135 . . . . . . 7 ((𝜑𝑥𝐼) → ∀𝑧𝑅 (𝑃𝑧 → (𝑧𝑆) ≠ ∅))
19 dfac14lem.s . . . . . . . . . . . 12 ((𝜑𝑥𝐼) → 𝑆𝑊)
20 snex 5428 . . . . . . . . . . . 12 {𝑃} ∈ V
21 unexg 7747 . . . . . . . . . . . 12 ((𝑆𝑊 ∧ {𝑃} ∈ V) → (𝑆 ∪ {𝑃}) ∈ V)
2219, 20, 21sylancl 584 . . . . . . . . . . 11 ((𝜑𝑥𝐼) → (𝑆 ∪ {𝑃}) ∈ V)
23 ssun2 4172 . . . . . . . . . . . 12 {𝑃} ⊆ (𝑆 ∪ {𝑃})
24 dfac14lem.p . . . . . . . . . . . . . 14 𝑃 = 𝒫 𝑆
25 uniexg 7741 . . . . . . . . . . . . . . 15 (𝑆𝑊 𝑆 ∈ V)
26 pwexg 5373 . . . . . . . . . . . . . . 15 ( 𝑆 ∈ V → 𝒫 𝑆 ∈ V)
2719, 25, 263syl 18 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐼) → 𝒫 𝑆 ∈ V)
2824, 27eqeltrid 2830 . . . . . . . . . . . . 13 ((𝜑𝑥𝐼) → 𝑃 ∈ V)
29 snidg 4658 . . . . . . . . . . . . 13 (𝑃 ∈ V → 𝑃 ∈ {𝑃})
3028, 29syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝐼) → 𝑃 ∈ {𝑃})
3123, 30sselid 3977 . . . . . . . . . . 11 ((𝜑𝑥𝐼) → 𝑃 ∈ (𝑆 ∪ {𝑃}))
32 epttop 22998 . . . . . . . . . . 11 (((𝑆 ∪ {𝑃}) ∈ V ∧ 𝑃 ∈ (𝑆 ∪ {𝑃})) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
3322, 31, 32syl2anc 582 . . . . . . . . . 10 ((𝜑𝑥𝐼) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
344, 33eqeltrid 2830 . . . . . . . . 9 ((𝜑𝑥𝐼) → 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
35 topontop 22901 . . . . . . . . 9 (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → 𝑅 ∈ Top)
3634, 35syl 17 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑅 ∈ Top)
37 toponuni 22902 . . . . . . . . . 10 (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → (𝑆 ∪ {𝑃}) = 𝑅)
3834, 37syl 17 . . . . . . . . 9 ((𝜑𝑥𝐼) → (𝑆 ∪ {𝑃}) = 𝑅)
399, 38sseqtrid 4032 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑆 𝑅)
4031, 38eleqtrd 2828 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑃 𝑅)
41 eqid 2726 . . . . . . . . 9 𝑅 = 𝑅
4241elcls 23063 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 𝑅𝑃 𝑅) → (𝑃 ∈ ((cls‘𝑅)‘𝑆) ↔ ∀𝑧𝑅 (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
4336, 39, 40, 42syl3anc 1368 . . . . . . 7 ((𝜑𝑥𝐼) → (𝑃 ∈ ((cls‘𝑅)‘𝑆) ↔ ∀𝑧𝑅 (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
4418, 43mpbird 256 . . . . . 6 ((𝜑𝑥𝐼) → 𝑃 ∈ ((cls‘𝑅)‘𝑆))
4544ralrimiva 3136 . . . . 5 (𝜑 → ∀𝑥𝐼 𝑃 ∈ ((cls‘𝑅)‘𝑆))
46 dfac14lem.i . . . . . 6 (𝜑𝐼𝑉)
47 mptelixpg 8954 . . . . . 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 256 . . . 4 (𝜑 → (𝑥𝐼𝑃) ∈ X𝑥𝐼 ((cls‘𝑅)‘𝑆))
5049ne0d 4336 . . 3 (𝜑X𝑥𝐼 ((cls‘𝑅)‘𝑆) ≠ ∅)
51 dfac14lem.c . . 3 (𝜑 → ((cls‘𝐽)‘X𝑥𝐼 𝑆) = X𝑥𝐼 ((cls‘𝑅)‘𝑆))
5234ralrimiva 3136 . . . . 5 (𝜑 → ∀𝑥𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
53 dfac14lem.j . . . . . 6 𝐽 = (∏t‘(𝑥𝐼𝑅))
5453pttopon 23586 . . . . 5 ((𝐼𝑉 ∧ ∀𝑥𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃}))) → 𝐽 ∈ (TopOn‘X𝑥𝐼 (𝑆 ∪ {𝑃})))
5546, 52, 54syl2anc 582 . . . 4 (𝜑𝐽 ∈ (TopOn‘X𝑥𝐼 (𝑆 ∪ {𝑃})))
56 topontop 22901 . . . 4 (𝐽 ∈ (TopOn‘X𝑥𝐼 (𝑆 ∪ {𝑃})) → 𝐽 ∈ Top)
57 cls0 23070 . . . 4 (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
5855, 56, 573syl 18 . . 3 (𝜑 → ((cls‘𝐽)‘∅) = ∅)
5950, 51, 583netr4d 3008 . 2 (𝜑 → ((cls‘𝐽)‘X𝑥𝐼 𝑆) ≠ ((cls‘𝐽)‘∅))
60 fveq2 6891 . . 3 (X𝑥𝐼 𝑆 = ∅ → ((cls‘𝐽)‘X𝑥𝐼 𝑆) = ((cls‘𝐽)‘∅))
6160necon3i 2963 . 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 205  wa 394   = wceq 1534  wcel 2099  wne 2930  wral 3051  {crab 3420  Vcvv 3463  cun 3945  cin 3946  wss 3947  c0 4323  𝒫 cpw 4598  {csn 4624   cuni 4906  cmpt 5227  cfv 6544  Xcixp 8916  tcpt 17446  Topctop 22881  TopOnctopon 22898  clsccl 23008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-int 4948  df-iun 4996  df-iin 4997  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7867  df-1o 8486  df-2o 8487  df-ixp 8917  df-en 8965  df-fin 8968  df-fi 9445  df-topgen 17451  df-pt 17452  df-top 22882  df-topon 22899  df-bases 22935  df-cld 23009  df-ntr 23010  df-cls 23011
This theorem is referenced by:  dfac14  23608
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