Theorem cvmfolem 35285

Description: Lemma for cvmfo 35306. (Contributed by Mario Carneiro, 13-Feb-2015.)

Hypotheses

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))))})
cvmseu.1	⊢ 𝐵 = ∪ 𝐶
cvmfolem.2	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
cvmfolem	⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹:𝐵–onto→𝑋)

Distinct variable groups:   𝑘,𝑠,𝑢,𝑣,𝐶   𝑘,𝐹,𝑠,𝑢,𝑣   𝑘,𝐽,𝑠,𝑢,𝑣   𝑣,𝐵

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

Proof of Theorem cvmfolem

Dummy variables 𝑡 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmcn 35268	. . 3 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
2		cvmseu.1	. . . 4 ⊢ 𝐵 = ∪ 𝐶
3		cvmfolem.2	. . . 4 ⊢ 𝑋 = ∪ 𝐽
4	2, 3	cnf 23255	. . 3 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶𝑋)
5	1, 4	syl 17	. 2 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹:𝐵⟶𝑋)
6		cvmcov.1	. . . . . 6 ⊢ 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, 3	cvmcov 35269	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥 ∈ 𝑋) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ (𝑆‘𝑧) ≠ ∅))
8	7	ex 412	. . . 4 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝑥 ∈ 𝑋 → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ (𝑆‘𝑧) ≠ ∅)))
9		n0 4352	. . . . . . 7 ⊢ ((𝑆‘𝑧) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝑆‘𝑧))
10	6	cvmsn0 35274	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (𝑆‘𝑧) → 𝑤 ≠ ∅)
11	10	ad2antll 729	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ (𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧))) → 𝑤 ≠ ∅)
12		n0 4352	. . . . . . . . . . 11 ⊢ (𝑤 ≠ ∅ ↔ ∃𝑡 𝑡 ∈ 𝑤)
13	11, 12	sylib 218	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ (𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧))) → ∃𝑡 𝑡 ∈ 𝑤)
14		simprlr 779	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑤 ∈ (𝑆‘𝑧))
15	6	cvmsss 35273	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (𝑆‘𝑧) → 𝑤 ⊆ 𝐶)
16	14, 15	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑤 ⊆ 𝐶)
17		simprr 772	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑡 ∈ 𝑤)
18	16, 17	sseldd 3983	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑡 ∈ 𝐶)
19		elssuni 4936	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ 𝐶 → 𝑡 ⊆ ∪ 𝐶)
20	18, 19	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑡 ⊆ ∪ 𝐶)
21	20, 2	sseqtrrdi 4024	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑡 ⊆ 𝐵)
22		simpll 766	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
23	6	cvmsf1o 35278	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑤 ∈ (𝑆‘𝑧) ∧ 𝑡 ∈ 𝑤) → (𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑧)
24	22, 14, 17, 23	syl3anc 1372	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → (𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑧)
25		f1ocnv 6859	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑧 → ◡(𝐹 ↾ 𝑡):𝑧–1-1-onto→𝑡)
26		f1of 6847	. . . . . . . . . . . . . . . 16 ⊢ (◡(𝐹 ↾ 𝑡):𝑧–1-1-onto→𝑡 → ◡(𝐹 ↾ 𝑡):𝑧⟶𝑡)
27	24, 25, 26	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → ◡(𝐹 ↾ 𝑡):𝑧⟶𝑡)
28		simprll 778	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑥 ∈ 𝑧)
29	27, 28	ffvelcdmd 7104	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → (◡(𝐹 ↾ 𝑡)‘𝑥) ∈ 𝑡)
30	21, 29	sseldd 3983	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → (◡(𝐹 ↾ 𝑡)‘𝑥) ∈ 𝐵)
31		f1ocnvfv2 7298	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑧 ∧ 𝑥 ∈ 𝑧) → ((𝐹 ↾ 𝑡)‘(◡(𝐹 ↾ 𝑡)‘𝑥)) = 𝑥)
32	24, 28, 31	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → ((𝐹 ↾ 𝑡)‘(◡(𝐹 ↾ 𝑡)‘𝑥)) = 𝑥)
33		fvres 6924	. . . . . . . . . . . . . . 15 ⊢ ((◡(𝐹 ↾ 𝑡)‘𝑥) ∈ 𝑡 → ((𝐹 ↾ 𝑡)‘(◡(𝐹 ↾ 𝑡)‘𝑥)) = (𝐹‘(◡(𝐹 ↾ 𝑡)‘𝑥)))
34	29, 33	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → ((𝐹 ↾ 𝑡)‘(◡(𝐹 ↾ 𝑡)‘𝑥)) = (𝐹‘(◡(𝐹 ↾ 𝑡)‘𝑥)))
35	32, 34	eqtr3d 2778	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑥 = (𝐹‘(◡(𝐹 ↾ 𝑡)‘𝑥)))
36		fveq2 6905	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (◡(𝐹 ↾ 𝑡)‘𝑥) → (𝐹‘𝑦) = (𝐹‘(◡(𝐹 ↾ 𝑡)‘𝑥)))
37	36	rspceeqv 3644	. . . . . . . . . . . . 13 ⊢ (((◡(𝐹 ↾ 𝑡)‘𝑥) ∈ 𝐵 ∧ 𝑥 = (𝐹‘(◡(𝐹 ↾ 𝑡)‘𝑥))) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))
38	30, 35, 37	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))
39	38	expr 456	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ (𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧))) → (𝑡 ∈ 𝑤 → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
40	39	exlimdv 1932	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ (𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧))) → (∃𝑡 𝑡 ∈ 𝑤 → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
41	13, 40	mpd 15	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ (𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧))) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))
42	41	expr 456	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ 𝑥 ∈ 𝑧) → (𝑤 ∈ (𝑆‘𝑧) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
43	42	exlimdv 1932	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ 𝑥 ∈ 𝑧) → (∃𝑤 𝑤 ∈ (𝑆‘𝑧) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
44	9, 43	biimtrid 242	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ 𝑥 ∈ 𝑧) → ((𝑆‘𝑧) ≠ ∅ → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
45	44	expimpd 453	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) → ((𝑥 ∈ 𝑧 ∧ (𝑆‘𝑧) ≠ ∅) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
46	45	rexlimdva 3154	. . . 4 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → (∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ (𝑆‘𝑧) ≠ ∅) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
47	8, 46	syld 47	. . 3 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝑥 ∈ 𝑋 → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
48	47	ralrimiv 3144	. 2 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))
49		dffo3 7121	. 2 ⊢ (𝐹:𝐵–onto→𝑋 ↔ (𝐹:𝐵⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
50	5, 48, 49	sylanbrc 583	1 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹:𝐵–onto→𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  {crab 3435   ∖ cdif 3947   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  𝒫 cpw 4599  {csn 4625  ∪ cuni 4906   ↦ cmpt 5224  ^◡ccnv 5683   ↾ cres 5686   “ cima 5687  ⟶wf 6556  –onto→wfo 6558  –1-1-onto→wf1o 6559  ‘cfv 6560  (class class class)co 7432   ↾_t crest 17466   Cn ccn 23233  Homeochmeo 23762   CovMap ccvm 35261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-map 8869  df-en 8987  df-fin 8990  df-fi 9452  df-rest 17468  df-topgen 17489  df-top 22901  df-topon 22918  df-bases 22954  df-cn 23236  df-hmeo 23764  df-cvm 35262

This theorem is referenced by:  cvmfo  35306