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Theorem dfac14lem 21622
Description: Lemma for dfac14 21623. 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 21621 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 2823 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑃𝑦𝑃𝑧))
2 eqeq1 2764 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦 = (𝑆 ∪ {𝑃}) ↔ 𝑧 = (𝑆 ∪ {𝑃})))
31, 2imbi12d 333 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝑃𝑦𝑦 = (𝑆 ∪ {𝑃})) ↔ (𝑃𝑧𝑧 = (𝑆 ∪ {𝑃}))))
4 dfac14lem.r . . . . . . . . . 10 𝑅 = {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))}
53, 4elrab2 3507 . . . . . . . . 9 (𝑧𝑅 ↔ (𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∧ (𝑃𝑧𝑧 = (𝑆 ∪ {𝑃}))))
6 dfac14lem.0 . . . . . . . . . . . . 13 ((𝜑𝑥𝐼) → 𝑆 ≠ ∅)
76adantr 472 . . . . . . . . . . . 12 (((𝜑𝑥𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → 𝑆 ≠ ∅)
8 ineq1 3950 . . . . . . . . . . . . . 14 (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧𝑆) = ((𝑆 ∪ {𝑃}) ∩ 𝑆))
9 ssun1 3919 . . . . . . . . . . . . . . 15 𝑆 ⊆ (𝑆 ∪ {𝑃})
10 sseqin2 3960 . . . . . . . . . . . . . . 15 (𝑆 ⊆ (𝑆 ∪ {𝑃}) ↔ ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆)
119, 10mpbi 220 . . . . . . . . . . . . . 14 ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆
128, 11syl6eq 2810 . . . . . . . . . . . . 13 (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧𝑆) = 𝑆)
1312neeq1d 2991 . . . . . . . . . . . 12 (𝑧 = (𝑆 ∪ {𝑃}) → ((𝑧𝑆) ≠ ∅ ↔ 𝑆 ≠ ∅))
147, 13syl5ibrcom 237 . . . . . . . . . . 11 (((𝜑𝑥𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧𝑆) ≠ ∅))
1514imim2d 57 . . . . . . . . . 10 (((𝜑𝑥𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → ((𝑃𝑧𝑧 = (𝑆 ∪ {𝑃})) → (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
1615expimpd 630 . . . . . . . . 9 ((𝜑𝑥𝐼) → ((𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∧ (𝑃𝑧𝑧 = (𝑆 ∪ {𝑃}))) → (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
175, 16syl5bi 232 . . . . . . . 8 ((𝜑𝑥𝐼) → (𝑧𝑅 → (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
1817ralrimiv 3103 . . . . . . 7 ((𝜑𝑥𝐼) → ∀𝑧𝑅 (𝑃𝑧 → (𝑧𝑆) ≠ ∅))
19 dfac14lem.s . . . . . . . . . . . 12 ((𝜑𝑥𝐼) → 𝑆𝑊)
20 snex 5057 . . . . . . . . . . . 12 {𝑃} ∈ V
21 unexg 7124 . . . . . . . . . . . 12 ((𝑆𝑊 ∧ {𝑃} ∈ V) → (𝑆 ∪ {𝑃}) ∈ V)
2219, 20, 21sylancl 697 . . . . . . . . . . 11 ((𝜑𝑥𝐼) → (𝑆 ∪ {𝑃}) ∈ V)
23 ssun2 3920 . . . . . . . . . . . 12 {𝑃} ⊆ (𝑆 ∪ {𝑃})
24 dfac14lem.p . . . . . . . . . . . . . 14 𝑃 = 𝒫 𝑆
25 uniexg 7120 . . . . . . . . . . . . . . 15 (𝑆𝑊 𝑆 ∈ V)
26 pwexg 4999 . . . . . . . . . . . . . . 15 ( 𝑆 ∈ V → 𝒫 𝑆 ∈ V)
2719, 25, 263syl 18 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐼) → 𝒫 𝑆 ∈ V)
2824, 27syl5eqel 2843 . . . . . . . . . . . . 13 ((𝜑𝑥𝐼) → 𝑃 ∈ V)
29 snidg 4351 . . . . . . . . . . . . 13 (𝑃 ∈ V → 𝑃 ∈ {𝑃})
3028, 29syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝐼) → 𝑃 ∈ {𝑃})
3123, 30sseldi 3742 . . . . . . . . . . 11 ((𝜑𝑥𝐼) → 𝑃 ∈ (𝑆 ∪ {𝑃}))
32 epttop 21015 . . . . . . . . . . 11 (((𝑆 ∪ {𝑃}) ∈ V ∧ 𝑃 ∈ (𝑆 ∪ {𝑃})) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
3322, 31, 32syl2anc 696 . . . . . . . . . 10 ((𝜑𝑥𝐼) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
344, 33syl5eqel 2843 . . . . . . . . 9 ((𝜑𝑥𝐼) → 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
35 topontop 20920 . . . . . . . . 9 (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → 𝑅 ∈ Top)
3634, 35syl 17 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑅 ∈ Top)
37 toponuni 20921 . . . . . . . . . 10 (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → (𝑆 ∪ {𝑃}) = 𝑅)
3834, 37syl 17 . . . . . . . . 9 ((𝜑𝑥𝐼) → (𝑆 ∪ {𝑃}) = 𝑅)
399, 38syl5sseq 3794 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑆 𝑅)
4031, 38eleqtrd 2841 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑃 𝑅)
41 eqid 2760 . . . . . . . . 9 𝑅 = 𝑅
4241elcls 21079 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 𝑅𝑃 𝑅) → (𝑃 ∈ ((cls‘𝑅)‘𝑆) ↔ ∀𝑧𝑅 (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
4336, 39, 40, 42syl3anc 1477 . . . . . . 7 ((𝜑𝑥𝐼) → (𝑃 ∈ ((cls‘𝑅)‘𝑆) ↔ ∀𝑧𝑅 (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
4418, 43mpbird 247 . . . . . 6 ((𝜑𝑥𝐼) → 𝑃 ∈ ((cls‘𝑅)‘𝑆))
4544ralrimiva 3104 . . . . 5 (𝜑 → ∀𝑥𝐼 𝑃 ∈ ((cls‘𝑅)‘𝑆))
46 dfac14lem.i . . . . . 6 (𝜑𝐼𝑉)
47 mptelixpg 8111 . . . . . 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 247 . . . 4 (𝜑 → (𝑥𝐼𝑃) ∈ X𝑥𝐼 ((cls‘𝑅)‘𝑆))
50 ne0i 4064 . . . 4 ((𝑥𝐼𝑃) ∈ X𝑥𝐼 ((cls‘𝑅)‘𝑆) → X𝑥𝐼 ((cls‘𝑅)‘𝑆) ≠ ∅)
5149, 50syl 17 . . 3 (𝜑X𝑥𝐼 ((cls‘𝑅)‘𝑆) ≠ ∅)
52 dfac14lem.c . . 3 (𝜑 → ((cls‘𝐽)‘X𝑥𝐼 𝑆) = X𝑥𝐼 ((cls‘𝑅)‘𝑆))
5334ralrimiva 3104 . . . . 5 (𝜑 → ∀𝑥𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
54 dfac14lem.j . . . . . 6 𝐽 = (∏t‘(𝑥𝐼𝑅))
5554pttopon 21601 . . . . 5 ((𝐼𝑉 ∧ ∀𝑥𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃}))) → 𝐽 ∈ (TopOn‘X𝑥𝐼 (𝑆 ∪ {𝑃})))
5646, 53, 55syl2anc 696 . . . 4 (𝜑𝐽 ∈ (TopOn‘X𝑥𝐼 (𝑆 ∪ {𝑃})))
57 topontop 20920 . . . 4 (𝐽 ∈ (TopOn‘X𝑥𝐼 (𝑆 ∪ {𝑃})) → 𝐽 ∈ Top)
58 cls0 21086 . . . 4 (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
5956, 57, 583syl 18 . . 3 (𝜑 → ((cls‘𝐽)‘∅) = ∅)
6051, 52, 593netr4d 3009 . 2 (𝜑 → ((cls‘𝐽)‘X𝑥𝐼 𝑆) ≠ ((cls‘𝐽)‘∅))
61 fveq2 6352 . . 3 (X𝑥𝐼 𝑆 = ∅ → ((cls‘𝐽)‘X𝑥𝐼 𝑆) = ((cls‘𝐽)‘∅))
6261necon3i 2964 . 2 (((cls‘𝐽)‘X𝑥𝐼 𝑆) ≠ ((cls‘𝐽)‘∅) → X𝑥𝐼 𝑆 ≠ ∅)
6360, 62syl 17 1 (𝜑X𝑥𝐼 𝑆 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  {crab 3054  Vcvv 3340  cun 3713  cin 3714  wss 3715  c0 4058  𝒫 cpw 4302  {csn 4321   cuni 4588  cmpt 4881  cfv 6049  Xcixp 8074  tcpt 16301  Topctop 20900  TopOnctopon 20917  clsccl 21024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-ixp 8075  df-en 8122  df-fin 8125  df-fi 8482  df-topgen 16306  df-pt 16307  df-top 20901  df-topon 20918  df-bases 20952  df-cld 21025  df-ntr 21026  df-cls 21027
This theorem is referenced by:  dfac14  21623
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