Theorem cvmfolem 31261

Description: Lemma for cvmfo 31282. (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 31244	. . 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 21050	. . 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 31245	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑥 ∈ 𝑋) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ (𝑆‘𝑧) ≠ ∅))
8	7	ex 450	. . . 4 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝑥 ∈ 𝑋 → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ (𝑆‘𝑧) ≠ ∅)))
9		n0 3931	. . . . . . 7 ⊢ ((𝑆‘𝑧) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝑆‘𝑧))
10	6	cvmsn0 31250	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (𝑆‘𝑧) → 𝑤 ≠ ∅)
11	10	ad2antll 765	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ (𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧))) → 𝑤 ≠ ∅)
12		n0 3931	. . . . . . . . . . 11 ⊢ (𝑤 ≠ ∅ ↔ ∃𝑡 𝑡 ∈ 𝑤)
13	11, 12	sylib 208	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ (𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧))) → ∃𝑡 𝑡 ∈ 𝑤)
14		simprlr 803	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑤 ∈ (𝑆‘𝑧))
15	6	cvmsss 31249	. . . . . . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑡 ∈ 𝑤)
18	16, 17	sseldd 3604	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑡 ∈ 𝐶)
19		elssuni 4467	. . . . . . . . . . . . . . . 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	syl6sseqr 3652	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑡 ⊆ 𝐵)
22		simpll 790	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
23	6	cvmsf1o 31254	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑤 ∈ (𝑆‘𝑧) ∧ 𝑡 ∈ 𝑤) → (𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑧)
24	22, 14, 17, 23	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → (𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑧)
25		f1ocnv 6149	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑧 → ◡(𝐹 ↾ 𝑡):𝑧–1-1-onto→𝑡)
26		f1of 6137	. . . . . . . . . . . . . . . 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 802	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑥 ∈ 𝑧)
29	27, 28	ffvelrnd 6360	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → (◡(𝐹 ↾ 𝑡)‘𝑥) ∈ 𝑡)
30	21, 29	sseldd 3604	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → (◡(𝐹 ↾ 𝑡)‘𝑥) ∈ 𝐵)
31		f1ocnvfv2 6533	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ↾ 𝑡):𝑡–1-1-onto→𝑧 ∧ 𝑥 ∈ 𝑧) → ((𝐹 ↾ 𝑡)‘(◡(𝐹 ↾ 𝑡)‘𝑥)) = 𝑥)
32	24, 28, 31	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → ((𝐹 ↾ 𝑡)‘(◡(𝐹 ↾ 𝑡)‘𝑥)) = 𝑥)
33		fvres 6207	. . . . . . . . . . . . . . 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 2658	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → 𝑥 = (𝐹‘(◡(𝐹 ↾ 𝑡)‘𝑥)))
36		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (◡(𝐹 ↾ 𝑡)‘𝑥) → (𝐹‘𝑦) = (𝐹‘(◡(𝐹 ↾ 𝑡)‘𝑥)))
37	36	eqeq2d 2632	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (◡(𝐹 ↾ 𝑡)‘𝑥) → (𝑥 = (𝐹‘𝑦) ↔ 𝑥 = (𝐹‘(◡(𝐹 ↾ 𝑡)‘𝑥))))
38	37	rspcev 3309	. . . . . . . . . . . . 13 ⊢ (((◡(𝐹 ↾ 𝑡)‘𝑥) ∈ 𝐵 ∧ 𝑥 = (𝐹‘(◡(𝐹 ↾ 𝑡)‘𝑥))) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))
39	30, 35, 38	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ ((𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧)) ∧ 𝑡 ∈ 𝑤)) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))
40	39	expr 643	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ (𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧))) → (𝑡 ∈ 𝑤 → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
41	40	exlimdv 1861	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ (𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧))) → (∃𝑡 𝑡 ∈ 𝑤 → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
42	13, 41	mpd 15	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ (𝑥 ∈ 𝑧 ∧ 𝑤 ∈ (𝑆‘𝑧))) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))
43	42	expr 643	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ 𝑥 ∈ 𝑧) → (𝑤 ∈ (𝑆‘𝑧) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
44	43	exlimdv 1861	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ 𝑥 ∈ 𝑧) → (∃𝑤 𝑤 ∈ (𝑆‘𝑧) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
45	9, 44	syl5bi 232	. . . . . 6 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) ∧ 𝑥 ∈ 𝑧) → ((𝑆‘𝑧) ≠ ∅ → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
46	45	expimpd 629	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑧 ∈ 𝐽) → ((𝑥 ∈ 𝑧 ∧ (𝑆‘𝑧) ≠ ∅) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
47	46	rexlimdva 3031	. . . 4 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → (∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ (𝑆‘𝑧) ≠ ∅) → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
48	8, 47	syld 47	. . 3 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝑥 ∈ 𝑋 → ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
49	48	ralrimiv 2965	. 2 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))
50		dffo3 6374	. 2 ⊢ (𝐹:𝐵–onto→𝑋 ↔ (𝐹:𝐵⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
51	5, 49, 50	sylanbrc 698	1 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹:𝐵–onto→𝑋)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436   ↦ cmpt 4729  ^◡ccnv 5113   ↾ cres 5116   “ cima 5117  ⟶wf 5884  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↾_t crest 16081   Cn ccn 21028  Homeochmeo 21556   CovMap ccvm 31237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-hmeo 21558  df-cvm 31238

This theorem is referenced by:  cvmfo  31282