Theorem cvmfolem 32530

Description: Lemma for cvmfo 32551. (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 32513	. . 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 21857	. . 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 32514	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥 ∈ 𝑋) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ (𝑆‘𝑧) ≠ ∅))
8	7	ex 415	. . . 4 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝑥 ∈ 𝑋 → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ (𝑆‘𝑧) ≠ ∅)))
9		n0 4313	. . . . . . 7 ⊢ ((𝑆‘𝑧) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝑆‘𝑧))
10	6	cvmsn0 32519	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (𝑆‘𝑧) → 𝑤 ≠ ∅)
11	10	ad2antll 727	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ (𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧))) → 𝑤 ≠ ∅)
12		n0 4313	. . . . . . . . . . 11 ⊢ (𝑤 ≠ ∅ ↔ ∃𝑡 𝑡 ∈ 𝑤)
13	11, 12	sylib 220	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ (𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧))) → ∃𝑡 𝑡 ∈ 𝑤)
14		simprlr 778	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑤 ∈ (𝑆‘𝑧))
15	6	cvmsss 32518	. . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑡 ∈ 𝑤)
18	16, 17	sseldd 3971	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑡 ∈ 𝐶)
19		elssuni 4871	. . . . . . . . . . . . . . . 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 4021	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑡 ⊆ 𝐵)
22		simpll 765	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
23	6	cvmsf1o 32523	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑤 ∈ (𝑆‘𝑧) ∧ 𝑡 ∈ 𝑤) → (𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑧)
24	22, 14, 17, 23	syl3anc 1367	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → (𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑧)
25		f1ocnv 6630	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑧 → ◡(𝐹 ↾ 𝑡):𝑧–1-1-onto→𝑡)
26		f1of 6618	. . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑥 ∈ 𝑧)
29	27, 28	ffvelrnd 6855	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → (◡(𝐹 ↾ 𝑡)‘𝑥) ∈ 𝑡)
30	21, 29	sseldd 3971	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → (◡(𝐹 ↾ 𝑡)‘𝑥) ∈ 𝐵)
31		f1ocnvfv2 7037	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑧 ∧ 𝑥 ∈ 𝑧) → ((𝐹 ↾ 𝑡)‘(◡(𝐹 ↾ 𝑡)‘𝑥)) = 𝑥)
32	24, 28, 31	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → ((𝐹 ↾ 𝑡)‘(◡(𝐹 ↾ 𝑡)‘𝑥)) = 𝑥)
33		fvres 6692	. . . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑥 = (𝐹‘(◡(𝐹 ↾ 𝑡)‘𝑥)))
36		fveq2 6673	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (◡(𝐹 ↾ 𝑡)‘𝑥) → (𝐹‘𝑦) = (𝐹‘(◡(𝐹 ↾ 𝑡)‘𝑥)))
37	36	rspceeqv 3641	. . . . . . . . . . . . 13 ⊢ (((◡(𝐹 ↾ 𝑡)‘𝑥) ∈ 𝐵 ∧ 𝑥 = (𝐹‘(◡(𝐹 ↾ 𝑡)‘𝑥))) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))
38	30, 35, 37	syl2anc 586	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))
39	38	expr 459	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ (𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧))) → (𝑡 ∈ 𝑤 → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
40	39	exlimdv 1933	. . . . . . . . . 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 459	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ 𝑥 ∈ 𝑧) → (𝑤 ∈ (𝑆‘𝑧) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
43	42	exlimdv 1933	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ 𝑥 ∈ 𝑧) → (∃𝑤 𝑤 ∈ (𝑆‘𝑧) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
44	9, 43	syl5bi 244	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ 𝑥 ∈ 𝑧) → ((𝑆‘𝑧) ≠ ∅ → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
45	44	expimpd 456	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) → ((𝑥 ∈ 𝑧 ∧ (𝑆‘𝑧) ≠ ∅) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
46	45	rexlimdva 3287	. . . 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 3184	. 2 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))
49		dffo3 6871	. 2 ⊢ (𝐹:𝐵–onto→𝑋 ↔ (𝐹:𝐵⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
50	5, 48, 49	sylanbrc 585	1 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹:𝐵–onto→𝑋)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536  ∃wex 1779   ∈ wcel 2113   ≠ wne 3019  ∀wral 3141  ∃wrex 3142  {crab 3145   ∖ cdif 3936   ∩ cin 3938   ⊆ wss 3939  ∅c0 4294  𝒫 cpw 4542  {csn 4570  ∪ cuni 4841   ↦ cmpt 5149  ^◡ccnv 5557   ↾ cres 5560   “ cima 5561  ⟶wf 6354  –onto→wfo 6356  –1-1-onto→wf1o 6357  ‘cfv 6358  (class class class)co 7159   ↾_t crest 16697   Cn ccn 21835  Homeochmeo 22364   CovMap ccvm 32506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-oadd 8109  df-er 8292  df-map 8411  df-en 8513  df-fin 8516  df-fi 8878  df-rest 16699  df-topgen 16720  df-top 21505  df-topon 21522  df-bases 21557  df-cn 21838  df-hmeo 22366  df-cvm 32507

This theorem is referenced by:  cvmfo  32551