Theorem dfac14lem 21700

Description: Lemma for dfac14 21701. 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 21699 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 2828	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ 𝑧))
2		eqeq1 2769	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑦 = (𝑆 ∪ {𝑃}) ↔ 𝑧 = (𝑆 ∪ {𝑃})))
3	1, 2	imbi12d 335	. . . . . . . . . 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 3523	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑅 ↔ (𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∧ (𝑃 ∈ 𝑧 → 𝑧 = (𝑆 ∪ {𝑃}))))
6		dfac14lem.0	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ≠ ∅)
7	6	adantr 472	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃})) → 𝑆 ≠ ∅)
8		ineq1 3969	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧 ∩ 𝑆) = ((𝑆 ∪ {𝑃}) ∩ 𝑆))
9		ssun1 3938	. . . . . . . . . . . . . . 15 ⊢ 𝑆 ⊆ (𝑆 ∪ {𝑃})
10		sseqin2 3979	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ⊆ (𝑆 ∪ {𝑃}) ↔ ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆)
11	9, 10	mpbi 221	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∪ {𝑃}) ∩ 𝑆) = 𝑆
12	8, 11	syl6eq 2815	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑆 ∪ {𝑃}) → (𝑧 ∩ 𝑆) = 𝑆)
13	12	neeq1d 2996	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑆 ∪ {𝑃}) → ((𝑧 ∩ 𝑆) ≠ ∅ ↔ 𝑆 ≠ ∅))
14	7, 13	syl5ibrcom 238	. . . . . . . . . . 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 445	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑧 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∧ (𝑃 ∈ 𝑧 → 𝑧 = (𝑆 ∪ {𝑃}))) → (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅)))
17	5, 16	syl5bi 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑧 ∈ 𝑅 → (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅)))
18	17	ralrimiv 3112	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∀𝑧 ∈ 𝑅 (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅))
19		dfac14lem.s	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ 𝑊)
20		snex 5064	. . . . . . . . . . . 12 ⊢ {𝑃} ∈ V
21		unexg 7157	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑊 ∧ {𝑃} ∈ V) → (𝑆 ∪ {𝑃}) ∈ V)
22	19, 20, 21	sylancl 580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆 ∪ {𝑃}) ∈ V)
23		ssun2 3939	. . . . . . . . . . . 12 ⊢ {𝑃} ⊆ (𝑆 ∪ {𝑃})
24		dfac14lem.p	. . . . . . . . . . . . . 14 ⊢ 𝑃 = 𝒫 ∪ 𝑆
25		uniexg 7153	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ 𝑊 → ∪ 𝑆 ∈ V)
26		pwexg 5014	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑆 ∈ V → 𝒫 ∪ 𝑆 ∈ V)
27	19, 25, 26	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝒫 ∪ 𝑆 ∈ V)
28	24, 27	syl5eqel 2848	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ V)
29		snidg 4364	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ V → 𝑃 ∈ {𝑃})
30	28, 29	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ {𝑃})
31	23, 30	sseldi 3759	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ (𝑆 ∪ {𝑃}))
32		epttop 21093	. . . . . . . . . . 11 ⊢ (((𝑆 ∪ {𝑃}) ∈ V ∧ 𝑃 ∈ (𝑆 ∪ {𝑃})) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃 ∈ 𝑦 → 𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
33	22, 31, 32	syl2anc 579	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃 ∈ 𝑦 → 𝑦 = (𝑆 ∪ {𝑃}))} ∈ (TopOn‘(𝑆 ∪ {𝑃})))
34	4, 33	syl5eqel 2848	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
35		topontop 20997	. . . . . . . . 9 ⊢ (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → 𝑅 ∈ Top)
36	34, 35	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ Top)
37		toponuni 20998	. . . . . . . . . 10 ⊢ (𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})) → (𝑆 ∪ {𝑃}) = ∪ 𝑅)
38	34, 37	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆 ∪ {𝑃}) = ∪ 𝑅)
39	9, 38	syl5sseq 3813	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ⊆ ∪ 𝑅)
40	31, 38	eleqtrd 2846	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ ∪ 𝑅)
41		eqid 2765	. . . . . . . . 9 ⊢ ∪ 𝑅 = ∪ 𝑅
42	41	elcls 21157	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅 ∧ 𝑃 ∈ ∪ 𝑅) → (𝑃 ∈ ((cls‘𝑅)‘𝑆) ↔ ∀𝑧 ∈ 𝑅 (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅)))
43	36, 39, 40, 42	syl3anc 1490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑃 ∈ ((cls‘𝑅)‘𝑆) ↔ ∀𝑧 ∈ 𝑅 (𝑃 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅)))
44	18, 43	mpbird 248	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑃 ∈ ((cls‘𝑅)‘𝑆))
45	44	ralrimiva 3113	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑃 ∈ ((cls‘𝑅)‘𝑆))
46		dfac14lem.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
47		mptelixpg 8150	. . . . . 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 248	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝑃) ∈ X𝑥 ∈ 𝐼 ((cls‘𝑅)‘𝑆))
50	49	ne0d 4086	. . 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 3113	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃})))
53		dfac14lem.j	. . . . . 6 ⊢ 𝐽 = (∏t‘(𝑥 ∈ 𝐼 ↦ 𝑅))
54	53	pttopon 21679	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐼 𝑅 ∈ (TopOn‘(𝑆 ∪ {𝑃}))) → 𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐼 (𝑆 ∪ {𝑃})))
55	46, 52, 54	syl2anc 579	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐼 (𝑆 ∪ {𝑃})))
56		topontop 20997	. . . 4 ⊢ (𝐽 ∈ (TopOn‘X𝑥 ∈ 𝐼 (𝑆 ∪ {𝑃})) → 𝐽 ∈ Top)
57		cls0 21164	. . . 4 ⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
58	55, 56, 57	3syl 18	. . 3 ⊢ (𝜑 → ((cls‘𝐽)‘∅) = ∅)
59	50, 51, 58	3netr4d 3014	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘X𝑥 ∈ 𝐼 𝑆) ≠ ((cls‘𝐽)‘∅))
60		fveq2 6375	. . 3 ⊢ (X𝑥 ∈ 𝐼 𝑆 = ∅ → ((cls‘𝐽)‘X𝑥 ∈ 𝐼 𝑆) = ((cls‘𝐽)‘∅))
61	60	necon3i 2969	. 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 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