Theorem dfac14lem 21230

Description: Lemma for dfac14 21231. 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 21229 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		eleq2 2677	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ 𝑧))
2		eqeq1 2614	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑦 = (𝑆 ∪ {𝑃}) ↔ 𝑧 = (𝑆 ∪ {𝑃})))
3	1, 2	imbi12d 333	. . . . . . . . . 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 3333	. . . . . . . . 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 3769	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧 ∩ 𝑆) = ((𝑆 ∪ {𝑃}) ∩ 𝑆))
9		ssun1 3738	. . . . . . . . . . . . . . 15 ⊢ 𝑆 ⊆ (𝑆 ∪ {𝑃})
10		sseqin2 3779	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ⊆ (𝑆 ∪ {𝑃}) ↔ ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆)
11	9, 10	mpbi 219	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆
12	8, 11	syl6eq 2660	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧 ∩ 𝑆) = 𝑆)
13	12	neeq1d 2841	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑆 ∪ {𝑃}) → ((𝑧 ∩ 𝑆) ≠ ∅ ↔ 𝑆 ≠ ∅))
14	7, 13	syl5ibrcom 236	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧 ∩ 𝑆) ≠ ∅))
15	14	imim2d 55	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → ((𝑃 ∈ 𝑧 → 𝑧 = (𝑆 ∪ {𝑃})) → (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅)))
16	15	expimpd 627	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∧ (𝑃 ∈ 𝑧 → 𝑧 = (𝑆 ∪ {𝑃}))) → (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅)))
17	5, 16	syl5bi 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑧 ∈ 𝑅 → (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅)))
18	17	ralrimiv 2948	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∀𝑧 ∈ 𝑅 (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅))
19		dfac14lem.s	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ 𝑊)
20		snex 4835	. . . . . . . . . . . 12 ⊢ {𝑃} ∈ V
21		unexg 6857	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑊 ∧ {𝑃} ∈ V) → (𝑆 ∪ {𝑃}) ∈ V)
22	19, 20, 21	sylancl 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆 ∪ {𝑃}) ∈ V)
23		ssun2 3739	. . . . . . . . . . . 12 ⊢ {𝑃} ⊆ (𝑆 ∪ {𝑃})
24		dfac14lem.p	. . . . . . . . . . . . . 14 ⊢ 𝑃 = 𝒫 ∪ 𝑆
25		uniexg 6853	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ 𝑊 → ∪ 𝑆 ∈ V)
26		pwexg 4776	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑆 ∈ V → 𝒫 ∪ 𝑆 ∈ V)
27	19, 25, 26	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝒫 ∪ 𝑆 ∈ V)
28	24, 27	syl5eqel 2692	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ V)
29		snidg 4153	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ V → 𝑃 ∈ {𝑃})
30	28, 29	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ {𝑃})
31	23, 30	sseldi 3566	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ (𝑆 ∪ {𝑃}))
32		epttop 20623	. . . . . . . . . . 11 ⊢ (((𝑆 ∪ {𝑃}) ∈ V ∧ 𝑃 ∈ (𝑆 ∪ {𝑃})) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃 ∈ 𝑦 → 𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
33	22, 31, 32	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃 ∈ 𝑦 → 𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
34	4, 33	syl5eqel 2692	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
35		topontop 20541	. . . . . . . . 9 ⊢ (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → 𝑅 ∈ Top)
36	34, 35	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ Top)
37		toponuni 20542	. . . . . . . . . 10 ⊢ (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → (𝑆 ∪ {𝑃}) = ∪ 𝑅)
38	34, 37	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆 ∪ {𝑃}) = ∪ 𝑅)
39	9, 38	syl5sseq 3616	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ⊆ ∪ 𝑅)
40	31, 38	eleqtrd 2690	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ ∪ 𝑅)
41		eqid 2610	. . . . . . . . 9 ⊢ ∪ 𝑅 = ∪ 𝑅
42	41	elcls 20687	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅 ∧ 𝑃 ∈ ∪ 𝑅) → (𝑃 ∈ ((cls‘𝑅)‘𝑆) ↔ ∀𝑧 ∈ 𝑅 (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅)))
43	36, 39, 40, 42	syl3anc 1318	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑃 ∈ ((cls‘𝑅)‘𝑆) ↔ ∀𝑧 ∈ 𝑅 (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅)))
44	18, 43	mpbird 246	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ ((cls‘𝑅)‘𝑆))
45	44	ralrimiva 2949	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑃 ∈ ((cls‘𝑅)‘𝑆))
46		dfac14lem.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
47		mptelixpg 7831	. . . . . 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 246	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝑃) ∈ X𝑥 ∈ 𝐼 ((cls‘𝑅)‘𝑆))
50		ne0i 3880	. . . 4 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝑃) ∈ X𝑥 ∈ 𝐼 ((cls‘𝑅)‘𝑆) → X𝑥 ∈ 𝐼 ((cls‘𝑅)‘𝑆) ≠ ∅)
51	49, 50	syl 17	. . 3 ⊢ (𝜑 → X𝑥 ∈ 𝐼 ((cls‘𝑅)‘𝑆) ≠ ∅)
52		dfac14lem.c	. . 3 ⊢ (𝜑 → ((cls‘𝐽)‘X𝑥 ∈ 𝐼 𝑆) = X𝑥 ∈ 𝐼 ((cls‘𝑅)‘𝑆))
53	34	ralrimiva 2949	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
54		dfac14lem.j	. . . . . 6 ⊢ 𝐽 = (∏t‘(𝑥 ∈ 𝐼 ↦ 𝑅))
55	54	pttopon 21209	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃}))) → 𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐼 (𝑆 ∪ {𝑃})))
56	46, 53, 55	syl2anc 691	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐼 (𝑆 ∪ {𝑃})))
57		topontop 20541	. . . 4 ⊢ (𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐼 (𝑆 ∪ {𝑃})) → 𝐽 ∈ Top)
58		cls0 20694	. . . 4 ⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
59	56, 57, 58	3syl 18	. . 3 ⊢ (𝜑 → ((cls‘𝐽)‘∅) = ∅)
60	51, 52, 59	3netr4d 2859	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘X𝑥 ∈ 𝐼 𝑆) ≠ ((cls‘𝐽)‘∅))
61		fveq2 6103	. . 3 ⊢ (X𝑥 ∈ 𝐼 𝑆 = ∅ → ((cls‘𝐽)‘X𝑥 ∈ 𝐼 𝑆) = ((cls‘𝐽)‘∅))
62	61	necon3i 2814	. 2 ⊢ (((cls‘𝐽)‘X𝑥 ∈ 𝐼 𝑆) ≠ ((cls‘𝐽)‘∅) → X𝑥 ∈ 𝐼 𝑆 ≠ ∅)
63	60, 62	syl 17	1 ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372   ↦ cmpt 4643  ‘cfv 5804  Xcixp 7794  ∏_tcpt 15922  Topctop 20517  TopOnctopon 20518  clsccl 20632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-ixp 7795  df-en 7842  df-fin 7845  df-fi 8200  df-topgen 15927  df-pt 15928  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-ntr 20634  df-cls 20635

This theorem is referenced by:  dfac14  21231