Theorem dfac14lem 23582

Description: Lemma for dfac14 23583. 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 23581 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 2820	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ 𝑧))
2		eqeq1 2740	. . . . . . . . . . 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 3637	. . . . . . . . 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 4153	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧 ∩ 𝑆) = ((𝑆 ∪ {𝑃}) ∩ 𝑆))
9		ssun1 4118	. . . . . . . . . . . . . . 15 ⊢ 𝑆 ⊆ (𝑆 ∪ {𝑃})
10		sseqin2 4163	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ⊆ (𝑆 ∪ {𝑃}) ↔ ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆)
11	9, 10	mpbi 230	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆
12	8, 11	eqtrdi 2787	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧 ∩ 𝑆) = 𝑆)
13	12	neeq1d 2991	. . . . . . . . . . . 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 3128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∀𝑧 ∈ 𝑅 (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅))
19		dfac14lem.s	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ 𝑊)
20		snex 5381	. . . . . . . . . . . 12 ⊢ {𝑃} ∈ V
21		unexg 7697	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑊 ∧ {𝑃} ∈ V) → (𝑆 ∪ {𝑃}) ∈ V)
22	19, 20, 21	sylancl 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆 ∪ {𝑃}) ∈ V)
23		ssun2 4119	. . . . . . . . . . . 12 ⊢ {𝑃} ⊆ (𝑆 ∪ {𝑃})
24		dfac14lem.p	. . . . . . . . . . . . . 14 ⊢ 𝑃 = 𝒫 ∪ 𝑆
25		uniexg 7694	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ 𝑊 → ∪ 𝑆 ∈ V)
26		pwexg 5320	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑆 ∈ V → 𝒫 ∪ 𝑆 ∈ V)
27	19, 25, 26	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝒫 ∪ 𝑆 ∈ V)
28	24, 27	eqeltrid 2840	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ V)
29		snidg 4604	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ V → 𝑃 ∈ {𝑃})
30	28, 29	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ {𝑃})
31	23, 30	sselid 3919	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ (𝑆 ∪ {𝑃}))
32		epttop 22974	. . . . . . . . . . 11 ⊢ (((𝑆 ∪ {𝑃}) ∈ V ∧ 𝑃 ∈ (𝑆 ∪ {𝑃})) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃 ∈ 𝑦 → 𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
33	22, 31, 32	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃 ∈ 𝑦 → 𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
34	4, 33	eqeltrid 2840	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
35		topontop 22878	. . . . . . . . 9 ⊢ (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → 𝑅 ∈ Top)
36	34, 35	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ Top)
37		toponuni 22879	. . . . . . . . . 10 ⊢ (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → (𝑆 ∪ {𝑃}) = ∪ 𝑅)
38	34, 37	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆 ∪ {𝑃}) = ∪ 𝑅)
39	9, 38	sseqtrid 3964	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ⊆ ∪ 𝑅)
40	31, 38	eleqtrd 2838	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ ∪ 𝑅)
41		eqid 2736	. . . . . . . . 9 ⊢ ∪ 𝑅 = ∪ 𝑅
42	41	elcls 23038	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅 ∧ 𝑃 ∈ ∪ 𝑅) → (𝑃 ∈ ((cls‘𝑅)‘𝑆) ↔ ∀𝑧 ∈ 𝑅 (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅)))
43	36, 39, 40, 42	syl3anc 1374	. . . . . . 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 3129	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑃 ∈ ((cls‘𝑅)‘𝑆))
46		dfac14lem.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
47		mptelixpg 8883	. . . . . 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 4282	. . 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 3129	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
53		dfac14lem.j	. . . . . 6 ⊢ 𝐽 = (∏t‘(𝑥 ∈ 𝐼 ↦ 𝑅))
54	53	pttopon 23561	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃}))) → 𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐼 (𝑆 ∪ {𝑃})))
55	46, 52, 54	syl2anc 585	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐼 (𝑆 ∪ {𝑃})))
56		topontop 22878	. . . 4 ⊢ (𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐼 (𝑆 ∪ {𝑃})) → 𝐽 ∈ Top)
57		cls0 23045	. . . 4 ⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
58	55, 56, 57	3syl 18	. . 3 ⊢ (𝜑 → ((cls‘𝐽)‘∅) = ∅)
59	50, 51, 58	3netr4d 3009	. 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 2964	. 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 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  {crab 3389  Vcvv 3429   ∪ cun 3887   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541  {csn 4567  ∪ cuni 4850   ↦ cmpt 5166  ‘cfv 6498  Xcixp 8845  ∏_tcpt 17401  Topctop 22858  TopOnctopon 22875  clsccl 22983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  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 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-om 7818  df-1o 8405  df-2o 8406  df-ixp 8846  df-en 8894  df-fin 8897  df-fi 9324  df-topgen 17406  df-pt 17407  df-top 22859  df-topon 22876  df-bases 22911  df-cld 22984  df-ntr 22985  df-cls 22986

This theorem is referenced by:  dfac14  23583