dyadmbllem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dyadmbllem		Structured version Visualization version GIF version

Theorem dyadmbllem 24668

Description: Lemma for dyadmbl 24669. (Contributed by Mario Carneiro, 26-Mar-2015.)

**Hypotheses**
Ref	Expression
dyadmbl.1	⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
dyadmbl.2	⊢ 𝐺 = {𝑧 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)}
dyadmbl.3	⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)

**Assertion**
Ref	Expression
dyadmbllem	⊢ (𝜑 → ∪ ([,] “ 𝐴) = ∪ ([,] “ 𝐺))

Distinct variable groups: 𝑥,𝑦 𝑧,𝑤,𝜑 𝑥,𝑤,𝑦,𝐴,𝑧 𝑧,𝐺 𝑤,𝐹,𝑥,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑥,𝑦) 𝐺(𝑥,𝑦,𝑤)

Proof of Theorem dyadmbllem Dummy variables 𝑎 𝑚 𝑡 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni2 4840	. . . 4 ⊢ (𝑎 ∈ ∪ ([,] “ 𝐴) ↔ ∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖)
2		iccf 13109	. . . . . . 7 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
3		ffn 6584	. . . . . . 7 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → [,] Fn (ℝ^* × ℝ^*))
4	2, 3	ax-mp 5	. . . . . 6 ⊢ [,] Fn (ℝ^* × ℝ^*)
5		dyadmbl.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)
6		dyadmbl.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
7	6	dyadf 24660	. . . . . . . . 9 ⊢ 𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ))
8		frn 6591	. . . . . . . . 9 ⊢ (𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
9	7, 8	ax-mp 5	. . . . . . . 8 ⊢ ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
10		inss2 4160	. . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
11		rexpssxrxp 10951	. . . . . . . . 9 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
12	10, 11	sstri 3926	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
13	9, 12	sstri 3926	. . . . . . 7 ⊢ ran 𝐹 ⊆ (ℝ^* × ℝ^*)
14	5, 13	sstrdi 3929	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (ℝ^* × ℝ^*))
15		eleq2 2827	. . . . . . 7 ⊢ (𝑖 = ([,]‘𝑡) → (𝑎 ∈ 𝑖 ↔ 𝑎 ∈ ([,]‘𝑡)))
16	15	rexima 7095	. . . . . 6 ⊢ (([,] Fn (ℝ^* × ℝ^) ∧ 𝐴 ⊆ (ℝ^ × ℝ^*)) → (∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖 ↔ ∃𝑡 ∈ 𝐴 𝑎 ∈ ([,]‘𝑡)))
17	4, 14, 16	sylancr 586	. . . . 5 ⊢ (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖 ↔ ∃𝑡 ∈ 𝐴 𝑎 ∈ ([,]‘𝑡)))
18		ssrab2 4009	. . . . . . . . 9 ⊢ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ 𝐴
19	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → 𝐴 ⊆ ran 𝐹)
20	18, 19	sstrid 3928	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ ran 𝐹)
21		simprl 767	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → 𝑡 ∈ 𝐴)
22		ssid 3939	. . . . . . . . . 10 ⊢ ([,]‘𝑡) ⊆ ([,]‘𝑡)
23		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑡 → ([,]‘𝑎) = ([,]‘𝑡))
24	23	sseq2d 3949	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑡 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑡)))
25	24	rspcev 3552	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑡)) → ∃𝑎 ∈ 𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
26	21, 22, 25	sylancl 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → ∃𝑎 ∈ 𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
27		rabn0 4316	. . . . . . . . 9 ⊢ ({𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅ ↔ ∃𝑎 ∈ 𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
28	26, 27	sylibr 233	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅)
29	6	dyadmax 24667	. . . . . . . 8 ⊢ (({𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ ran 𝐹 ∧ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅) → ∃𝑚 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
30	20, 28, 29	syl2anc 583	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → ∃𝑚 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
31		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑚 → ([,]‘𝑎) = ([,]‘𝑚))
32	31	sseq2d 3949	. . . . . . . . . 10 ⊢ (𝑎 = 𝑚 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
33	32	elrab 3617	. . . . . . . . 9 ⊢ (𝑚 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
34		simprlr 776	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ([,]‘𝑡) ⊆ ([,]‘𝑚))
35		simplrr 774	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ([,]‘𝑡))
36	34, 35	sseldd 3918	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ([,]‘𝑚))
37		simprll 775	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚 ∈ 𝐴)
38		fveq2 6756	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑤 → ([,]‘𝑎) = ([,]‘𝑤))
39	38	sseq2d 3949	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑤 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
40	39	elrab 3617	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑤 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
41	40	imbi1i 349	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ ((𝑤 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
42		impexp 450	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ (𝑤 ∈ 𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
43	41, 42	bitri 274	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ (𝑤 ∈ 𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
44		impexp 450	. . . . . . . . . . . . . . . . . 18 ⊢ (((([,]‘𝑡) ⊆ ([,]‘𝑤) ∧ ([,]‘𝑚) ⊆ ([,]‘𝑤)) → 𝑚 = 𝑤) ↔ (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
45		sstr2 3924	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (([,]‘𝑡) ⊆ ([,]‘𝑚) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
46	45	ad2antll 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
47	46	ancrd 551	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → (([,]‘𝑡) ⊆ ([,]‘𝑤) ∧ ([,]‘𝑚) ⊆ ([,]‘𝑤))))
48	47	imim1d 82	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (((([,]‘𝑡) ⊆ ([,]‘𝑤) ∧ ([,]‘𝑚) ⊆ ([,]‘𝑤)) → 𝑚 = 𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
49	44, 48	syl5bir 242	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → ((([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
50	49	imim2d 57	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → ((𝑤 ∈ 𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → (𝑤 ∈ 𝐴 → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
51	43, 50	syl5bi 241	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → ((𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) → (𝑤 ∈ 𝐴 → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
52	51	ralimdv2 3101	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → ∀𝑤 ∈ 𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
53	52	impr 454	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ∀𝑤 ∈ 𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
54		fveq2 6756	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑚 → ([,]‘𝑧) = ([,]‘𝑚))
55	54	sseq1d 3948	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑚 → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘𝑚) ⊆ ([,]‘𝑤)))
56		equequ1 2029	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑚 → (𝑧 = 𝑤 ↔ 𝑚 = 𝑤))
57	55, 56	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑚 → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
58	57	ralbidv 3120	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑚 → (∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤 ∈ 𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
59		dyadmbl.2	. . . . . . . . . . . . . 14 ⊢ 𝐺 = {𝑧 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)}
60	58, 59	elrab2 3620	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝐺 ↔ (𝑚 ∈ 𝐴 ∧ ∀𝑤 ∈ 𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
61	37, 53, 60	sylanbrc 582	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚 ∈ 𝐺)
62		ffun 6587	. . . . . . . . . . . . . 14 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → Fun [,])
63	2, 62	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun [,]
64	59	ssrab3 4011	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 ⊆ 𝐴
65	64, 14	sstrid 3928	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ⊆ (ℝ^* × ℝ^*))
66	2	fdmi 6596	. . . . . . . . . . . . . . 15 ⊢ dom [,] = (ℝ^* × ℝ^*)
67	65, 66	sseqtrrdi 3968	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ⊆ dom [,])
68	67	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝐺 ⊆ dom [,])
69		funfvima2 7089	. . . . . . . . . . . . 13 ⊢ ((Fun [,] ∧ 𝐺 ⊆ dom [,]) → (𝑚 ∈ 𝐺 → ([,]‘𝑚) ∈ ([,] “ 𝐺)))
70	63, 68, 69	sylancr 586	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → (𝑚 ∈ 𝐺 → ([,]‘𝑚) ∈ ([,] “ 𝐺)))
71	61, 70	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ([,]‘𝑚) ∈ ([,] “ 𝐺))
72		elunii 4841	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ([,]‘𝑚) ∧ ([,]‘𝑚) ∈ ([,] “ 𝐺)) → 𝑎 ∈ ∪ ([,] “ 𝐺))
73	36, 71, 72	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ∪ ([,] “ 𝐺))
74	73	exp32 420	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) → (∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ∈ ∪ ([,] “ 𝐺))))
75	33, 74	syl5bi 241	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → (𝑚 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ∈ ∪ ([,] “ 𝐺))))
76	75	rexlimdv 3211	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → (∃𝑚 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ∈ ∪ ([,] “ 𝐺)))
77	30, 76	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → 𝑎 ∈ ∪ ([,] “ 𝐺))
78	77	rexlimdvaa 3213	. . . . 5 ⊢ (𝜑 → (∃𝑡 ∈ 𝐴 𝑎 ∈ ([,]‘𝑡) → 𝑎 ∈ ∪ ([,] “ 𝐺)))
79	17, 78	sylbid 239	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖 → 𝑎 ∈ ∪ ([,] “ 𝐺)))
80	1, 79	syl5bi 241	. . 3 ⊢ (𝜑 → (𝑎 ∈ ∪ ([,] “ 𝐴) → 𝑎 ∈ ∪ ([,] “ 𝐺)))
81	80	ssrdv 3923	. 2 ⊢ (𝜑 → ∪ ([,] “ 𝐴) ⊆ ∪ ([,] “ 𝐺))
82		imass2 5999	. . . 4 ⊢ (𝐺 ⊆ 𝐴 → ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
83	64, 82	ax-mp 5	. . 3 ⊢ ([,] “ 𝐺) ⊆ ([,] “ 𝐴)
84		uniss 4844	. . 3 ⊢ (([,] “ 𝐺) ⊆ ([,] “ 𝐴) → ∪ ([,] “ 𝐺) ⊆ ∪ ([,] “ 𝐴))
85	83, 84	mp1i 13	. 2 ⊢ (𝜑 → ∪ ([,] “ 𝐺) ⊆ ∪ ([,] “ 𝐴))
86	81, 85	eqssd 3934	1 ⊢ (𝜑 → ∪ ([,] “ 𝐴) = ∪ ([,] “ 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 {crab 3067 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 𝒫 cpw 4530 ⟨cop 4564 ∪ cuni 4836 × cxp 5578 dom cdm 5580 ran crn 5581 “ cima 5583 Fun wfun 6412 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 ℝcr 10801 1c1 10803 + caddc 10805 ℝ^*cxr 10939 ≤ cle 10941 / cdiv 11562 2c2 11958 ℕ₀cn0 12163 ℤcz 12249 [,]cicc 13011 ↑cexp 13710

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-rest 17050 df-topgen 17071 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-top 21951 df-topon 21968 df-bases 22004 df-cmp 22446 df-ovol 24533

This theorem is referenced by: dyadmbl 24669 mblfinlem1 35741 mblfinlem2 35742

W3C validator