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Theorem dfac14lem 21700
Description: Lemma for dfac14 21701. 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 21699 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 2828 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑃𝑦𝑃𝑧))
2 eqeq1 2769 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦 = (𝑆 ∪ {𝑃}) ↔ 𝑧 = (𝑆 ∪ {𝑃})))
31, 2imbi12d 335 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝑃𝑦𝑦 = (𝑆 ∪ {𝑃})) ↔ (𝑃𝑧𝑧 = (𝑆 ∪ {𝑃}))))
4 dfac14lem.r . . . . . . . . . 10 𝑅 = {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))}
53, 4elrab2 3523 . . . . . . . . 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 3969 . . . . . . . . . . . . . 14 (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧𝑆) = ((𝑆 ∪ {𝑃}) ∩ 𝑆))
9 ssun1 3938 . . . . . . . . . . . . . . 15 𝑆 ⊆ (𝑆 ∪ {𝑃})
10 sseqin2 3979 . . . . . . . . . . . . . . 15 (𝑆 ⊆ (𝑆 ∪ {𝑃}) ↔ ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆)
119, 10mpbi 221 . . . . . . . . . . . . . 14 ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆
128, 11syl6eq 2815 . . . . . . . . . . . . 13 (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧𝑆) = 𝑆)
1312neeq1d 2996 . . . . . . . . . . . 12 (𝑧 = (𝑆 ∪ {𝑃}) → ((𝑧𝑆) ≠ ∅ ↔ 𝑆 ≠ ∅))
147, 13syl5ibrcom 238 . . . . . . . . . . 11 (((𝜑𝑥𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧𝑆) ≠ ∅))
1514imim2d 57 . . . . . . . . . 10 (((𝜑𝑥𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → ((𝑃𝑧𝑧 = (𝑆 ∪ {𝑃})) → (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
1615expimpd 445 . . . . . . . . 9 ((𝜑𝑥𝐼) → ((𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∧ (𝑃𝑧𝑧 = (𝑆 ∪ {𝑃}))) → (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
175, 16syl5bi 233 . . . . . . . 8 ((𝜑𝑥𝐼) → (𝑧𝑅 → (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
1817ralrimiv 3112 . . . . . . 7 ((𝜑𝑥𝐼) → ∀𝑧𝑅 (𝑃𝑧 → (𝑧𝑆) ≠ ∅))
19 dfac14lem.s . . . . . . . . . . . 12 ((𝜑𝑥𝐼) → 𝑆𝑊)
20 snex 5064 . . . . . . . . . . . 12 {𝑃} ∈ V
21 unexg 7157 . . . . . . . . . . . 12 ((𝑆𝑊 ∧ {𝑃} ∈ V) → (𝑆 ∪ {𝑃}) ∈ V)
2219, 20, 21sylancl 580 . . . . . . . . . . 11 ((𝜑𝑥𝐼) → (𝑆 ∪ {𝑃}) ∈ V)
23 ssun2 3939 . . . . . . . . . . . 12 {𝑃} ⊆ (𝑆 ∪ {𝑃})
24 dfac14lem.p . . . . . . . . . . . . . 14 𝑃 = 𝒫 𝑆
25 uniexg 7153 . . . . . . . . . . . . . . 15 (𝑆𝑊 𝑆 ∈ V)
26 pwexg 5014 . . . . . . . . . . . . . . 15 ( 𝑆 ∈ V → 𝒫 𝑆 ∈ V)
2719, 25, 263syl 18 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐼) → 𝒫 𝑆 ∈ V)
2824, 27syl5eqel 2848 . . . . . . . . . . . . 13 ((𝜑𝑥𝐼) → 𝑃 ∈ V)
29 snidg 4364 . . . . . . . . . . . . 13 (𝑃 ∈ V → 𝑃 ∈ {𝑃})
3028, 29syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝐼) → 𝑃 ∈ {𝑃})
3123, 30sseldi 3759 . . . . . . . . . . 11 ((𝜑𝑥𝐼) → 𝑃 ∈ (𝑆 ∪ {𝑃}))
32 epttop 21093 . . . . . . . . . . 11 (((𝑆 ∪ {𝑃}) ∈ V ∧ 𝑃 ∈ (𝑆 ∪ {𝑃})) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
3322, 31, 32syl2anc 579 . . . . . . . . . 10 ((𝜑𝑥𝐼) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
344, 33syl5eqel 2848 . . . . . . . . 9 ((𝜑𝑥𝐼) → 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
35 topontop 20997 . . . . . . . . 9 (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → 𝑅 ∈ Top)
3634, 35syl 17 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑅 ∈ Top)
37 toponuni 20998 . . . . . . . . . 10 (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → (𝑆 ∪ {𝑃}) = 𝑅)
3834, 37syl 17 . . . . . . . . 9 ((𝜑𝑥𝐼) → (𝑆 ∪ {𝑃}) = 𝑅)
399, 38syl5sseq 3813 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑆 𝑅)
4031, 38eleqtrd 2846 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑃 𝑅)
41 eqid 2765 . . . . . . . . 9 𝑅 = 𝑅
4241elcls 21157 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 𝑅𝑃 𝑅) → (𝑃 ∈ ((cls‘𝑅)‘𝑆) ↔ ∀𝑧𝑅 (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
4336, 39, 40, 42syl3anc 1490 . . . . . . 7 ((𝜑𝑥𝐼) → (𝑃 ∈ ((cls‘𝑅)‘𝑆) ↔ ∀𝑧𝑅 (𝑃𝑧 → (𝑧𝑆) ≠ ∅)))
4418, 43mpbird 248 . . . . . 6 ((𝜑𝑥𝐼) → 𝑃 ∈ ((cls‘𝑅)‘𝑆))
4544ralrimiva 3113 . . . . 5 (𝜑 → ∀𝑥𝐼 𝑃 ∈ ((cls‘𝑅)‘𝑆))
46 dfac14lem.i . . . . . 6 (𝜑𝐼𝑉)
47 mptelixpg 8150 . . . . . 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 248 . . . 4 (𝜑 → (𝑥𝐼𝑃) ∈ X𝑥𝐼 ((cls‘𝑅)‘𝑆))
5049ne0d 4086 . . 3 (𝜑X𝑥𝐼 ((cls‘𝑅)‘𝑆) ≠ ∅)
51 dfac14lem.c . . 3 (𝜑 → ((cls‘𝐽)‘X𝑥𝐼 𝑆) = X𝑥𝐼 ((cls‘𝑅)‘𝑆))
5234ralrimiva 3113 . . . . 5 (𝜑 → ∀𝑥𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
53 dfac14lem.j . . . . . 6 𝐽 = (∏t‘(𝑥𝐼𝑅))
5453pttopon 21679 . . . . 5 ((𝐼𝑉 ∧ ∀𝑥𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃}))) → 𝐽 ∈ (TopOn‘X𝑥𝐼 (𝑆 ∪ {𝑃})))
5546, 52, 54syl2anc 579 . . . 4 (𝜑𝐽 ∈ (TopOn‘X𝑥𝐼 (𝑆 ∪ {𝑃})))
56 topontop 20997 . . . 4 (𝐽 ∈ (TopOn‘X𝑥𝐼 (𝑆 ∪ {𝑃})) → 𝐽 ∈ Top)
57 cls0 21164 . . . 4 (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
5855, 56, 573syl 18 . . 3 (𝜑 → ((cls‘𝐽)‘∅) = ∅)
5950, 51, 583netr4d 3014 . 2 (𝜑 → ((cls‘𝐽)‘X𝑥𝐼 𝑆) ≠ ((cls‘𝐽)‘∅))
60 fveq2 6375 . . 3 (X𝑥𝐼 𝑆 = ∅ → ((cls‘𝐽)‘X𝑥𝐼 𝑆) = ((cls‘𝐽)‘∅))
6160necon3i 2969 . 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 197  wa 384   = wceq 1652  wcel 2155  wne 2937  wral 3055  {crab 3059  Vcvv 3350  cun 3730  cin 3731  wss 3732  c0 4079  𝒫 cpw 4315  {csn 4334   cuni 4594  cmpt 4888  cfv 6068  Xcixp 8113  tcpt 16367  Topctop 20977  TopOnctopon 20994  clsccl 21102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-ixp 8114  df-en 8161  df-fin 8164  df-fi 8524  df-topgen 16372  df-pt 16373  df-top 20978  df-topon 20995  df-bases 21030  df-cld 21103  df-ntr 21104  df-cls 21105
This theorem is referenced by:  dfac14  21701
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