Theorem dfac14lem 23537

Description: Lemma for dfac14 23538. By equipping 𝑆 ∪ {𝑃} for some 𝑃 ∉ 𝑆 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 23536 to extract an element of the closure of X𝑘 ∈ 𝐼𝑆. (Contributed by Mario Carneiro, 2-Sep-2015.)

Hypotheses

Ref	Expression
dfac14lem.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
dfac14lem.s	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ 𝑊)
dfac14lem.0	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ≠ ∅)
dfac14lem.p	⊢ 𝑃 = 𝒫 ∪ 𝑆
dfac14lem.r	⊢ 𝑅 = {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃 ∈ 𝑦 → 𝑦 = (𝑆 ∪ {𝑃}))}
dfac14lem.j	⊢ 𝐽 = (∏t‘(𝑥 ∈ 𝐼 ↦ 𝑅))
dfac14lem.c	⊢ (𝜑 → ((cls‘𝐽)‘X𝑥 ∈ 𝐼 𝑆) = X𝑥 ∈ 𝐼 ((cls‘𝑅)‘𝑆))

Assertion

Ref	Expression
dfac14lem	⊢ (𝜑 → X𝑥 ∈ 𝐼 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.

Step	Hyp	Ref	Expression
1		eleq2w 2812	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ 𝑧))
2		eqeq1 2733	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑦 = (𝑆 ∪ {𝑃}) ↔ 𝑧 = (𝑆 ∪ {𝑃})))
3	1, 2	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝑃 ∈ 𝑦 → 𝑦 = (𝑆 ∪ {𝑃})) ↔ (𝑃 ∈ 𝑧 → 𝑧 = (𝑆 ∪ {𝑃}))))
4		dfac14lem.r	. . . . . . . . . 10 ⊢ 𝑅 = {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃 ∈ 𝑦 → 𝑦 = (𝑆 ∪ {𝑃}))}
5	3, 4	elrab2 3659	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑅 ↔ (𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∧ (𝑃 ∈ 𝑧 → 𝑧 = (𝑆 ∪ {𝑃}))))
6		dfac14lem.0	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ≠ ∅)
7	6	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → 𝑆 ≠ ∅)
8		ineq1 4172	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧 ∩ 𝑆) = ((𝑆 ∪ {𝑃}) ∩ 𝑆))
9		ssun1 4137	. . . . . . . . . . . . . . 15 ⊢ 𝑆 ⊆ (𝑆 ∪ {𝑃})
10		sseqin2 4182	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ⊆ (𝑆 ∪ {𝑃}) ↔ ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆)
11	9, 10	mpbi 230	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆
12	8, 11	eqtrdi 2780	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧 ∩ 𝑆) = 𝑆)
13	12	neeq1d 2984	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑆 ∪ {𝑃}) → ((𝑧 ∩ 𝑆) ≠ ∅ ↔ 𝑆 ≠ ∅))
14	7, 13	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧 ∩ 𝑆) ≠ ∅))
15	14	imim2d 57	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → ((𝑃 ∈ 𝑧 → 𝑧 = (𝑆 ∪ {𝑃})) → (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅)))
16	15	expimpd 453	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∧ (𝑃 ∈ 𝑧 → 𝑧 = (𝑆 ∪ {𝑃}))) → (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅)))
17	5, 16	biimtrid 242	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑧 ∈ 𝑅 → (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅)))
18	17	ralrimiv 3124	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∀𝑧 ∈ 𝑅 (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅))
19		dfac14lem.s	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ 𝑊)
20		snex 5386	. . . . . . . . . . . 12 ⊢ {𝑃} ∈ V
21		unexg 7699	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑊 ∧ {𝑃} ∈ V) → (𝑆 ∪ {𝑃}) ∈ V)
22	19, 20, 21	sylancl 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆 ∪ {𝑃}) ∈ V)
23		ssun2 4138	. . . . . . . . . . . 12 ⊢ {𝑃} ⊆ (𝑆 ∪ {𝑃})
24		dfac14lem.p	. . . . . . . . . . . . . 14 ⊢ 𝑃 = 𝒫 ∪ 𝑆
25		uniexg 7696	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ 𝑊 → ∪ 𝑆 ∈ V)
26		pwexg 5328	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑆 ∈ V → 𝒫 ∪ 𝑆 ∈ V)
27	19, 25, 26	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝒫 ∪ 𝑆 ∈ V)
28	24, 27	eqeltrid 2832	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ V)
29		snidg 4620	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ V → 𝑃 ∈ {𝑃})
30	28, 29	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ {𝑃})
31	23, 30	sselid 3941	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ (𝑆 ∪ {𝑃}))
32		epttop 22929	. . . . . . . . . . 11 ⊢ (((𝑆 ∪ {𝑃}) ∈ V ∧ 𝑃 ∈ (𝑆 ∪ {𝑃})) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃 ∈ 𝑦 → 𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
33	22, 31, 32	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃 ∈ 𝑦 → 𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
34	4, 33	eqeltrid 2832	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
35		topontop 22833	. . . . . . . . 9 ⊢ (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → 𝑅 ∈ Top)
36	34, 35	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ Top)
37		toponuni 22834	. . . . . . . . . 10 ⊢ (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → (𝑆 ∪ {𝑃}) = ∪ 𝑅)
38	34, 37	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆 ∪ {𝑃}) = ∪ 𝑅)
39	9, 38	sseqtrid 3986	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ⊆ ∪ 𝑅)
40	31, 38	eleqtrd 2830	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ ∪ 𝑅)
41		eqid 2729	. . . . . . . . 9 ⊢ ∪ 𝑅 = ∪ 𝑅
42	41	elcls 22993	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅 ∧ 𝑃 ∈ ∪ 𝑅) → (𝑃 ∈ ((cls‘𝑅)‘𝑆) ↔ ∀𝑧 ∈ 𝑅 (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅)))
43	36, 39, 40, 42	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑃 ∈ ((cls‘𝑅)‘𝑆) ↔ ∀𝑧 ∈ 𝑅 (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅)))
44	18, 43	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ ((cls‘𝑅)‘𝑆))
45	44	ralrimiva 3125	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑃 ∈ ((cls‘𝑅)‘𝑆))
46		dfac14lem.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
47		mptelixpg 8885	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → ((𝑥 ∈ 𝐼 ↦ 𝑃) ∈ X𝑥 ∈ 𝐼 ((cls‘𝑅)‘𝑆) ↔ ∀𝑥 ∈ 𝐼 𝑃 ∈ ((cls‘𝑅)‘𝑆)))
48	46, 47	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ 𝑃) ∈ X𝑥 ∈ 𝐼 ((cls‘𝑅)‘𝑆) ↔ ∀𝑥 ∈ 𝐼 𝑃 ∈ ((cls‘𝑅)‘𝑆)))
49	45, 48	mpbird 257	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝑃) ∈ X𝑥 ∈ 𝐼 ((cls‘𝑅)‘𝑆))
50	49	ne0d 4301	. . 3 ⊢ (𝜑 → X𝑥 ∈ 𝐼 ((cls‘𝑅)‘𝑆) ≠ ∅)
51		dfac14lem.c	. . 3 ⊢ (𝜑 → ((cls‘𝐽)‘X𝑥 ∈ 𝐼 𝑆) = X𝑥 ∈ 𝐼 ((cls‘𝑅)‘𝑆))
52	34	ralrimiva 3125	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
53		dfac14lem.j	. . . . . 6 ⊢ 𝐽 = (∏t‘(𝑥 ∈ 𝐼 ↦ 𝑅))
54	53	pttopon 23516	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃}))) → 𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐼 (𝑆 ∪ {𝑃})))
55	46, 52, 54	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐼 (𝑆 ∪ {𝑃})))
56		topontop 22833	. . . 4 ⊢ (𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐼 (𝑆 ∪ {𝑃})) → 𝐽 ∈ Top)
57		cls0 23000	. . . 4 ⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
58	55, 56, 57	3syl 18	. . 3 ⊢ (𝜑 → ((cls‘𝐽)‘∅) = ∅)
59	50, 51, 58	3netr4d 3002	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘X𝑥 ∈ 𝐼 𝑆) ≠ ((cls‘𝐽)‘∅))
60		fveq2 6840	. . 3 ⊢ (X𝑥 ∈ 𝐼 𝑆 = ∅ → ((cls‘𝐽)‘X𝑥 ∈ 𝐼 𝑆) = ((cls‘𝐽)‘∅))
61	60	necon3i 2957	. 2 ⊢ (((cls‘𝐽)‘X𝑥 ∈ 𝐼 𝑆) ≠ ((cls‘𝐽)‘∅) → X𝑥 ∈ 𝐼 𝑆 ≠ ∅)
62	59, 61	syl 17	1 ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  {crab 3402  Vcvv 3444   ∪ cun 3909   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4559  {csn 4585  ∪ cuni 4867   ↦ cmpt 5183  ‘cfv 6499  Xcixp 8847  ∏_tcpt 17377  Topctop 22813  TopOnctopon 22830  clsccl 22938

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7823  df-1o 8411  df-2o 8412  df-ixp 8848  df-en 8896  df-fin 8899  df-fi 9338  df-topgen 17382  df-pt 17383  df-top 22814  df-topon 22831  df-bases 22866  df-cld 22939  df-ntr 22940  df-cls 22941

This theorem is referenced by:  dfac14  23538