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Theorem dfac14lem 21932
Description: Lemma for dfac14 21933. 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 21931 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 2849 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑃𝑦𝑃𝑧))
2 eqeq1 2782 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦 = (𝑆 ∪ {𝑃}) ↔ 𝑧 = (𝑆 ∪ {𝑃})))
31, 2imbi12d 337 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝑃𝑦𝑦 = (𝑆 ∪ {𝑃})) ↔ (𝑃𝑧𝑧 = (𝑆 ∪ {𝑃}))))
4 dfac14lem.r . . . . . . . . . 10 𝑅 = {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))}
53, 4elrab2 3599 . . . . . . . . 9 (𝑧𝑅 ↔ (𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∧ (𝑃𝑧𝑧 = (𝑆 ∪ {𝑃}))))
6 dfac14lem.0 . . . . . . . . . . . . 13 ((𝜑𝑥𝐼) → 𝑆 ≠ ∅)
76adantr 473 . . . . . . . . . . . 12 (((𝜑𝑥𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → 𝑆 ≠ ∅)
8 ineq1 4070 . . . . . . . . . . . . . 14 (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧𝑆) = ((𝑆 ∪ {𝑃}) ∩ 𝑆))
9 ssun1 4039 . . . . . . . . . . . . . . 15 𝑆 ⊆ (𝑆 ∪ {𝑃})
10 sseqin2 4081 . . . . . . . . . . . . . . 15 (𝑆 ⊆ (𝑆 ∪ {𝑃}) ↔ ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆)
119, 10mpbi 222 . . . . . . . . . . . . . 14 ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆
128, 11syl6eq 2830 . . . . . . . . . . . . 13 (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧𝑆) = 𝑆)
1312neeq1d 3026 . . . . . . . . . . . 12 (𝑧 = (𝑆 ∪ {𝑃}) → ((𝑧𝑆) ≠ ∅ ↔ 𝑆 ≠ ∅))
147, 13syl5ibrcom 239 . . . . . . . . . . 11 (((𝜑𝑥𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧𝑆) ≠ ∅))
1514imim2d 57 . . . . . . . . . 10 (((𝜑𝑥𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → ((𝑃𝑧𝑧 = (𝑆 ∪ {𝑃})) → (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
1615expimpd 446 . . . . . . . . 9 ((𝜑𝑥𝐼) → ((𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∧ (𝑃𝑧𝑧 = (𝑆 ∪ {𝑃}))) → (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
175, 16syl5bi 234 . . . . . . . 8 ((𝜑𝑥𝐼) → (𝑧𝑅 → (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
1817ralrimiv 3131 . . . . . . 7 ((𝜑𝑥𝐼) → ∀𝑧𝑅 (𝑃𝑧 → (𝑧𝑆) ≠ ∅))
19 dfac14lem.s . . . . . . . . . . . 12 ((𝜑𝑥𝐼) → 𝑆𝑊)
20 snex 5189 . . . . . . . . . . . 12 {𝑃} ∈ V
21 unexg 7291 . . . . . . . . . . . 12 ((𝑆𝑊 ∧ {𝑃} ∈ V) → (𝑆 ∪ {𝑃}) ∈ V)
2219, 20, 21sylancl 577 . . . . . . . . . . 11 ((𝜑𝑥𝐼) → (𝑆 ∪ {𝑃}) ∈ V)
23 ssun2 4040 . . . . . . . . . . . 12 {𝑃} ⊆ (𝑆 ∪ {𝑃})
24 dfac14lem.p . . . . . . . . . . . . . 14 𝑃 = 𝒫 𝑆
25 uniexg 7287 . . . . . . . . . . . . . . 15 (𝑆𝑊 𝑆 ∈ V)
26 pwexg 5133 . . . . . . . . . . . . . . 15 ( 𝑆 ∈ V → 𝒫 𝑆 ∈ V)
2719, 25, 263syl 18 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐼) → 𝒫 𝑆 ∈ V)
2824, 27syl5eqel 2870 . . . . . . . . . . . . 13 ((𝜑𝑥𝐼) → 𝑃 ∈ V)
29 snidg 4472 . . . . . . . . . . . . 13 (𝑃 ∈ V → 𝑃 ∈ {𝑃})
3028, 29syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝐼) → 𝑃 ∈ {𝑃})
3123, 30sseldi 3858 . . . . . . . . . . 11 ((𝜑𝑥𝐼) → 𝑃 ∈ (𝑆 ∪ {𝑃}))
32 epttop 21324 . . . . . . . . . . 11 (((𝑆 ∪ {𝑃}) ∈ V ∧ 𝑃 ∈ (𝑆 ∪ {𝑃})) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
3322, 31, 32syl2anc 576 . . . . . . . . . 10 ((𝜑𝑥𝐼) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
344, 33syl5eqel 2870 . . . . . . . . 9 ((𝜑𝑥𝐼) → 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
35 topontop 21228 . . . . . . . . 9 (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → 𝑅 ∈ Top)
3634, 35syl 17 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑅 ∈ Top)
37 toponuni 21229 . . . . . . . . . 10 (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → (𝑆 ∪ {𝑃}) = 𝑅)
3834, 37syl 17 . . . . . . . . 9 ((𝜑𝑥𝐼) → (𝑆 ∪ {𝑃}) = 𝑅)
399, 38syl5sseq 3911 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑆 𝑅)
4031, 38eleqtrd 2868 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑃 𝑅)
41 eqid 2778 . . . . . . . . 9 𝑅 = 𝑅
4241elcls 21388 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 𝑅𝑃 𝑅) → (𝑃 ∈ ((cls‘𝑅)‘𝑆) ↔ ∀𝑧𝑅 (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
4336, 39, 40, 42syl3anc 1351 . . . . . . 7 ((𝜑𝑥𝐼) → (𝑃 ∈ ((cls‘𝑅)‘𝑆) ↔ ∀𝑧𝑅 (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
4418, 43mpbird 249 . . . . . 6 ((𝜑𝑥𝐼) → 𝑃 ∈ ((cls‘𝑅)‘𝑆))
4544ralrimiva 3132 . . . . 5 (𝜑 → ∀𝑥𝐼 𝑃 ∈ ((cls‘𝑅)‘𝑆))
46 dfac14lem.i . . . . . 6 (𝜑𝐼𝑉)
47 mptelixpg 8298 . . . . . 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 249 . . . 4 (𝜑 → (𝑥𝐼𝑃) ∈ X𝑥𝐼 ((cls‘𝑅)‘𝑆))
5049ne0d 4189 . . 3 (𝜑X𝑥𝐼 ((cls‘𝑅)‘𝑆) ≠ ∅)
51 dfac14lem.c . . 3 (𝜑 → ((cls‘𝐽)‘X𝑥𝐼 𝑆) = X𝑥𝐼 ((cls‘𝑅)‘𝑆))
5234ralrimiva 3132 . . . . 5 (𝜑 → ∀𝑥𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
53 dfac14lem.j . . . . . 6 𝐽 = (∏t‘(𝑥𝐼𝑅))
5453pttopon 21911 . . . . 5 ((𝐼𝑉 ∧ ∀𝑥𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃}))) → 𝐽 ∈ (TopOn‘X𝑥𝐼 (𝑆 ∪ {𝑃})))
5546, 52, 54syl2anc 576 . . . 4 (𝜑𝐽 ∈ (TopOn‘X𝑥𝐼 (𝑆 ∪ {𝑃})))
56 topontop 21228 . . . 4 (𝐽 ∈ (TopOn‘X𝑥𝐼 (𝑆 ∪ {𝑃})) → 𝐽 ∈ Top)
57 cls0 21395 . . . 4 (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
5855, 56, 573syl 18 . . 3 (𝜑 → ((cls‘𝐽)‘∅) = ∅)
5950, 51, 583netr4d 3044 . 2 (𝜑 → ((cls‘𝐽)‘X𝑥𝐼 𝑆) ≠ ((cls‘𝐽)‘∅))
60 fveq2 6501 . . 3 (X𝑥𝐼 𝑆 = ∅ → ((cls‘𝐽)‘X𝑥𝐼 𝑆) = ((cls‘𝐽)‘∅))
6160necon3i 2999 . 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 198  wa 387   = wceq 1507  wcel 2050  wne 2967  wral 3088  {crab 3092  Vcvv 3415  cun 3829  cin 3830  wss 3831  c0 4180  𝒫 cpw 4423  {csn 4442   cuni 4713  cmpt 5009  cfv 6190  Xcixp 8261  tcpt 16571  Topctop 21208  TopOnctopon 21225  clsccl 21333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-int 4751  df-iun 4795  df-iin 4796  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-ov 6981  df-oprab 6982  df-mpo 6983  df-om 7399  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-1o 7907  df-oadd 7911  df-er 8091  df-ixp 8262  df-en 8309  df-fin 8312  df-fi 8672  df-topgen 16576  df-pt 16577  df-top 21209  df-topon 21226  df-bases 21261  df-cld 21334  df-ntr 21335  df-cls 21336
This theorem is referenced by:  dfac14  21933
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