Theorem cvmsf1o 35485

Description: 𝐹, localized to an element of an even covering of 𝑈, is a bijection. (Contributed by Mario Carneiro, 14-Feb-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
cvmsf1o	⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝑈)

Distinct variable groups:   𝑘,𝑠,𝑢,𝑣,𝐶   𝑘,𝐹,𝑠,𝑢,𝑣   𝑘,𝐽,𝑠,𝑢,𝑣   𝑈,𝑘,𝑠,𝑢,𝑣   𝑇,𝑠,𝑢,𝑣   𝑢,𝐴,𝑣

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

Proof of Theorem cvmsf1o

Step	Hyp	Ref	Expression
1		cvmtop1 35473	. . . . 5 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
2	1	3ad2ant1 1134	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐶 ∈ Top)
3		eqid 2737	. . . . 5 ⊢ ∪ 𝐶 = ∪ 𝐶
4	3	toptopon 22873	. . . 4 ⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘∪ 𝐶))
5	2, 4	sylib 218	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐶 ∈ (TopOn‘∪ 𝐶))
6		cvmcov.1	. . . . . . 7 ⊢ 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	6	cvmsss 35480	. . . . . 6 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑇 ⊆ 𝐶)
8	7	3ad2ant2 1135	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑇 ⊆ 𝐶)
9		simp3 1139	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝑇)
10	8, 9	sseldd 3936	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝐶)
11		elssuni 4896	. . . 4 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ⊆ ∪ 𝐶)
12	10, 11	syl 17	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ ∪ 𝐶)
13		resttopon 23117	. . 3 ⊢ ((𝐶 ∈ (TopOn‘∪ 𝐶) ∧ 𝐴 ⊆ ∪ 𝐶) → (𝐶 ↾t 𝐴) ∈ (TopOn‘𝐴))
14	5, 12, 13	syl2anc 585	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐶 ↾t 𝐴) ∈ (TopOn‘𝐴))
15		cvmtop2 35474	. . . . 5 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
16	15	3ad2ant1 1134	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐽 ∈ Top)
17		eqid 2737	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
18	17	toptopon 22873	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
19	16, 18	sylib 218	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐽 ∈ (TopOn‘∪ 𝐽))
20	6	cvmsrcl 35477	. . . . 5 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑈 ∈ 𝐽)
21	20	3ad2ant2 1135	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑈 ∈ 𝐽)
22		elssuni 4896	. . . 4 ⊢ (𝑈 ∈ 𝐽 → 𝑈 ⊆ ∪ 𝐽)
23	21, 22	syl 17	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑈 ⊆ ∪ 𝐽)
24		resttopon 23117	. . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑈 ⊆ ∪ 𝐽) → (𝐽 ↾t 𝑈) ∈ (TopOn‘𝑈))
25	19, 23, 24	syl2anc 585	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐽 ↾t 𝑈) ∈ (TopOn‘𝑈))
26	6	cvmshmeo 35484	. . 3 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐹 ↾ 𝐴) ∈ ((𝐶 ↾t 𝐴)Homeo(𝐽 ↾t 𝑈)))
27	26	3adant1 1131	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐹 ↾ 𝐴) ∈ ((𝐶 ↾t 𝐴)Homeo(𝐽 ↾t 𝑈)))
28		hmeof1o2 23719	. 2 ⊢ (((𝐶 ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ (𝐽 ↾t 𝑈) ∈ (TopOn‘𝑈) ∧ (𝐹 ↾ 𝐴) ∈ ((𝐶 ↾t 𝐴)Homeo(𝐽 ↾t 𝑈))) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝑈)
29	14, 25, 27, 28	syl3anc 1374	1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3401   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  {csn 4582  ∪ cuni 4865   ↦ cmpt 5181  ^◡ccnv 5631   ↾ cres 5634   “ cima 5635  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368   ↾_t crest 17352  Topctop 22849  TopOnctopon 22866  Homeochmeo 23709   CovMap ccvm 35468

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-map 8777  df-en 8896  df-fin 8899  df-fi 9326  df-rest 17354  df-topgen 17375  df-top 22850  df-topon 22867  df-bases 22902  df-cn 23183  df-hmeo 23711  df-cvm 35469

This theorem is referenced by:  cvmsss2  35487  cvmfolem  35492  cvmliftmolem1  35494  cvmliftlem6  35503  cvmliftlem9  35506  cvmlift2lem9a  35516