Theorem cvmcov2 35457

Description: The covering map property can be restricted to an open subset. (Contributed by Mario Carneiro, 7-Jul-2015.)

Hypothesis

Ref	Expression
cvmcov.1	⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})

Assertion

Ref	Expression
cvmcov2	⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑥 ∈ 𝒫 𝑈(𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅))

Distinct variable groups:   𝑘,𝑠,𝑢,𝑣,𝑥,𝐶   𝑘,𝐹,𝑠,𝑢,𝑣,𝑥   𝑃,𝑘,𝑥   𝑘,𝐽,𝑠,𝑢,𝑣,𝑥   𝑥,𝑆   𝑈,𝑘,𝑠,𝑢,𝑣,𝑥

Allowed substitution hints:   𝑃(𝑣,𝑢,𝑠)   𝑆(𝑣,𝑢,𝑘,𝑠)

Proof of Theorem cvmcov2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1137	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → 𝐹 ∈ (𝐶 CovMap 𝐽))
2		simp3 1139	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → 𝑃 ∈ 𝑈)
3		simp2 1138	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → 𝑈 ∈ 𝐽)
4		elunii 4855	. . . 4 ⊢ ((𝑃 ∈ 𝑈 ∧ 𝑈 ∈ 𝐽) → 𝑃 ∈ ∪ 𝐽)
5	2, 3, 4	syl2anc 585	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → 𝑃 ∈ ∪ 𝐽)
6		cvmcov.1	. . . 4 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})
7		eqid 2736	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
8	6, 7	cvmcov 35445	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃 ∈ ∪ 𝐽) → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))
9	1, 5, 8	syl2anc 585	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))
10		inss2 4178	. . . . 5 ⊢ (𝑦 ∩ 𝑈) ⊆ 𝑈
11		vex 3433	. . . . . . 7 ⊢ 𝑦 ∈ V
12	11	inex1 5258	. . . . . 6 ⊢ (𝑦 ∩ 𝑈) ∈ V
13	12	elpw 4545	. . . . 5 ⊢ ((𝑦 ∩ 𝑈) ∈ 𝒫 𝑈 ↔ (𝑦 ∩ 𝑈) ⊆ 𝑈)
14	10, 13	mpbir 231	. . . 4 ⊢ (𝑦 ∩ 𝑈) ∈ 𝒫 𝑈
15	14	a1i 11	. . 3 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → (𝑦 ∩ 𝑈) ∈ 𝒫 𝑈)
16		simprrl 781	. . . 4 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → 𝑃 ∈ 𝑦)
17	2	adantr 480	. . . 4 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → 𝑃 ∈ 𝑈)
18	16, 17	elind 4140	. . 3 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → 𝑃 ∈ (𝑦 ∩ 𝑈))
19		simprrr 782	. . . 4 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → (𝑆‘𝑦) ≠ ∅)
20	1	adantr 480	. . . . 5 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
21		cvmtop2 35443	. . . . . . 7 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
22	20, 21	syl 17	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → 𝐽 ∈ Top)
23		simprl 771	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → 𝑦 ∈ 𝐽)
24	3	adantr 480	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → 𝑈 ∈ 𝐽)
25		inopn 22864	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽 ∧ 𝑈 ∈ 𝐽) → (𝑦 ∩ 𝑈) ∈ 𝐽)
26	22, 23, 24, 25	syl3anc 1374	. . . . 5 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → (𝑦 ∩ 𝑈) ∈ 𝐽)
27		inss1 4177	. . . . . 6 ⊢ (𝑦 ∩ 𝑈) ⊆ 𝑦
28	27	a1i 11	. . . . 5 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → (𝑦 ∩ 𝑈) ⊆ 𝑦)
29	6	cvmsss2 35456	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑦 ∩ 𝑈) ∈ 𝐽 ∧ (𝑦 ∩ 𝑈) ⊆ 𝑦) → ((𝑆‘𝑦) ≠ ∅ → (𝑆‘(𝑦 ∩ 𝑈)) ≠ ∅))
30	20, 26, 28, 29	syl3anc 1374	. . . 4 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → ((𝑆‘𝑦) ≠ ∅ → (𝑆‘(𝑦 ∩ 𝑈)) ≠ ∅))
31	19, 30	mpd 15	. . 3 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → (𝑆‘(𝑦 ∩ 𝑈)) ≠ ∅)
32		eleq2 2825	. . . . 5 ⊢ (𝑥 = (𝑦 ∩ 𝑈) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑦 ∩ 𝑈)))
33		fveq2 6840	. . . . . 6 ⊢ (𝑥 = (𝑦 ∩ 𝑈) → (𝑆‘𝑥) = (𝑆‘(𝑦 ∩ 𝑈)))
34	33	neeq1d 2991	. . . . 5 ⊢ (𝑥 = (𝑦 ∩ 𝑈) → ((𝑆‘𝑥) ≠ ∅ ↔ (𝑆‘(𝑦 ∩ 𝑈)) ≠ ∅))
35	32, 34	anbi12d 633	. . . 4 ⊢ (𝑥 = (𝑦 ∩ 𝑈) → ((𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅) ↔ (𝑃 ∈ (𝑦 ∩ 𝑈) ∧ (𝑆‘(𝑦 ∩ 𝑈)) ≠ ∅)))
36	35	rspcev 3564	. . 3 ⊢ (((𝑦 ∩ 𝑈) ∈ 𝒫 𝑈 ∧ (𝑃 ∈ (𝑦 ∩ 𝑈) ∧ (𝑆‘(𝑦 ∩ 𝑈)) ≠ ∅)) → ∃𝑥 ∈ 𝒫 𝑈(𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅))
37	15, 18, 31, 36	syl12anc 837	. 2 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) ∧ (𝑦 ∈ 𝐽 ∧ (𝑃 ∈ 𝑦 ∧ (𝑆‘𝑦) ≠ ∅))) → ∃𝑥 ∈ 𝒫 𝑈(𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅))
38	9, 37	rexlimddv 3144	1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑥 ∈ 𝒫 𝑈(𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  {crab 3389   ∖ cdif 3886   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541  {csn 4567  ∪ cuni 4850   ↦ cmpt 5166  ^◡ccnv 5630   ↾ cres 5633   “ cima 5634  ‘cfv 6498  (class class class)co 7367   ↾_t crest 17383  Topctop 22858  Homeochmeo 23718   CovMap ccvm 35437

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-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-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-map 8775  df-en 8894  df-fin 8897  df-fi 9324  df-rest 17385  df-topgen 17406  df-top 22859  df-topon 22876  df-bases 22911  df-cn 23192  df-hmeo 23720  df-cvm 35438

This theorem is referenced by: (None)