Theorem cvmsi 35252

Description: One direction of cvmsval 35253. (Contributed by Mario Carneiro, 13-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
cvmsi	⊢ (𝑇 ∈ (𝑆‘𝑈) → (𝑈 ∈ 𝐽 ∧ (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 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

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

Proof of Theorem cvmsi

Step	Hyp	Ref	Expression
1		cvmcov.1	. . 3 ⊢ 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))))})
2	1	cvmsrcl 35251	. 2 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑈 ∈ 𝐽)
3		imaeq2 6027	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑈 → (◡𝐹 “ 𝑘) = (◡𝐹 “ 𝑈))
4	3	eqeq2d 2740	. . . . . . . . . 10 ⊢ (𝑘 = 𝑈 → (∪ 𝑠 = (◡𝐹 “ 𝑘) ↔ ∪ 𝑠 = (◡𝐹 “ 𝑈)))
5		oveq2 7395	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑈 → (𝐽 ↾t 𝑘) = (𝐽 ↾t 𝑈))
6	5	oveq2d 7403	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑈 → ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)) = ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))
7	6	eleq2d 2814	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑈 → ((𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)) ↔ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))
8	7	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑈 → ((∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))) ↔ (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))
9	8	ralbidv 3156	. . . . . . . . . 10 ⊢ (𝑘 = 𝑈 → (∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))) ↔ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))
10	4, 9	anbi12d 632	. . . . . . . . 9 ⊢ (𝑘 = 𝑈 → ((∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))) ↔ (∪ 𝑠 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))))
11	10	rabbidv 3413	. . . . . . . 8 ⊢ (𝑘 = 𝑈 → {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))} = {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))})
12	11, 1	fvmptss2 6994	. . . . . . 7 ⊢ (𝑆‘𝑈) ⊆ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))}
13	12	sseli 3942	. . . . . 6 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑇 ∈ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))})
14		unieq 4882	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → ∪ 𝑠 = ∪ 𝑇)
15	14	eqeq1d 2731	. . . . . . . 8 ⊢ (𝑠 = 𝑇 → (∪ 𝑠 = (◡𝐹 “ 𝑈) ↔ ∪ 𝑇 = (◡𝐹 “ 𝑈)))
16		difeq1 4082	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑇 → (𝑠 ∖ {𝑢}) = (𝑇 ∖ {𝑢}))
17	16	raleqdv 3299	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ↔ ∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅))
18	17	anbi1d 631	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → ((∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))) ↔ (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))
19	18	raleqbi1dv 3311	. . . . . . . 8 ⊢ (𝑠 = 𝑇 → (∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))) ↔ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))
20	15, 19	anbi12d 632	. . . . . . 7 ⊢ (𝑠 = 𝑇 → ((∪ 𝑠 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))) ↔ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))))
21	20	elrab 3659	. . . . . 6 ⊢ (𝑇 ∈ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))} ↔ (𝑇 ∈ (𝒫 𝐶 ∖ {∅}) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))))
22	13, 21	sylib 218	. . . . 5 ⊢ (𝑇 ∈ (𝑆‘𝑈) → (𝑇 ∈ (𝒫 𝐶 ∖ {∅}) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))))
23	22	simpld 494	. . . 4 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑇 ∈ (𝒫 𝐶 ∖ {∅}))
24		eldifsn 4750	. . . 4 ⊢ (𝑇 ∈ (𝒫 𝐶 ∖ {∅}) ↔ (𝑇 ∈ 𝒫 𝐶 ∧ 𝑇 ≠ ∅))
25	23, 24	sylib 218	. . 3 ⊢ (𝑇 ∈ (𝑆‘𝑈) → (𝑇 ∈ 𝒫 𝐶 ∧ 𝑇 ≠ ∅))
26		elpwi 4570	. . . 4 ⊢ (𝑇 ∈ 𝒫 𝐶 → 𝑇 ⊆ 𝐶)
27	26	anim1i 615	. . 3 ⊢ ((𝑇 ∈ 𝒫 𝐶 ∧ 𝑇 ≠ ∅) → (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅))
28	25, 27	syl 17	. 2 ⊢ (𝑇 ∈ (𝑆‘𝑈) → (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅))
29	22	simprd 495	. 2 ⊢ (𝑇 ∈ (𝑆‘𝑈) → (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))
30	2, 28, 29	3jca 1128	1 ⊢ (𝑇 ∈ (𝑆‘𝑈) → (𝑈 ∈ 𝐽 ∧ (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  {crab 3405   ∖ cdif 3911   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  {csn 4589  ∪ cuni 4871   ↦ cmpt 5188  ^◡ccnv 5637   ↾ cres 5640   “ cima 5641  ‘cfv 6511  (class class class)co 7387   ↾_t crest 17383  Homeochmeo 23640

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390

This theorem is referenced by:  cvmsval  35253  cvmsss  35254  cvmsn0  35255  cvmsuni  35256  cvmsdisj  35257  cvmshmeo  35258