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 25758

Description: Lemma for dyadmbl 25759. (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 4876	. . . 4 ⊢ (𝑎 ∈ ∪ ([,] “ 𝐴) ↔ ∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖)
2		iccf 13470	. . . . . . 7 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
3		ffn 6705	. . . . . . 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 25750	. . . . . . . . 9 ⊢ 𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ))
8		frn 6713	. . . . . . . . 9 ⊢ (𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
9	7, 8	ax-mp 5	. . . . . . . 8 ⊢ ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
10		inss2 4190	. . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
11		rexpssxrxp 11249	. . . . . . . . 9 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
12	10, 11	sstri 3946	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
13	9, 12	sstri 3946	. . . . . . 7 ⊢ ran 𝐹 ⊆ (ℝ^* × ℝ^*)
14	5, 13	sstrdi 3949	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (ℝ^* × ℝ^*))
15		eleq2 2852	. . . . . . 7 ⊢ (𝑖 = ([,]‘𝑡) → (𝑎 ∈ 𝑖 ↔ 𝑎 ∈ ([,]‘𝑡)))
16	15	rexima 7236	. . . . . 6 ⊢ (([,] Fn (ℝ^* × ℝ^) ∧ 𝐴 ⊆ (ℝ^ × ℝ^*)) → (∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖 ↔ ∃𝑡 ∈ 𝐴 𝑎 ∈ ([,]‘𝑡)))
17	4, 14, 16	sylancr 598	. . . . 5 ⊢ (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖 ↔ ∃𝑡 ∈ 𝐴 𝑎 ∈ ([,]‘𝑡)))
18		ssrab2 4034	. . . . . . . . 9 ⊢ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ 𝐴
19	5	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → 𝐴 ⊆ ran 𝐹)
20	18, 19	sstrid 3948	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ ran 𝐹)
21		simprl 782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → 𝑡 ∈ 𝐴)
22		ssid 3959	. . . . . . . . . 10 ⊢ ([,]‘𝑡) ⊆ ([,]‘𝑡)
23		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑡 → ([,]‘𝑎) = ([,]‘𝑡))
24	23	sseq2d 3969	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑡 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑡)))
25	24	rspcev 3581	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑡)) → ∃𝑎 ∈ 𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
26	21, 22, 25	sylancl 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → ∃𝑎 ∈ 𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
27		rabn0 4346	. . . . . . . . 9 ⊢ ({𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅ ↔ ∃𝑎 ∈ 𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
28	26, 27	sylibr 237	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅)
29	6	dyadmax 25757	. . . . . . . 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 595	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → ∃𝑚 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
31		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑚 → ([,]‘𝑎) = ([,]‘𝑚))
32	31	sseq2d 3969	. . . . . . . . . 10 ⊢ (𝑎 = 𝑚 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
33	32	elrab 3650	. . . . . . . . 9 ⊢ (𝑚 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
34		simprlr 791	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ([,]‘𝑡) ⊆ ([,]‘𝑚))
35		simplrr 789	. . . . . . . . . . . 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 3938	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ([,]‘𝑚))
37		simprll 790	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚 ∈ 𝐴)
38		fveq2 6881	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑤 → ([,]‘𝑎) = ([,]‘𝑤))
39	38	sseq2d 3969	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑤 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
40	39	elrab 3650	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑤 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
41	40	imbi1i 352	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ ((𝑤 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
42		impexp 455	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ (𝑤 ∈ 𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
43	41, 42	bitri 278	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ (𝑤 ∈ 𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
44		impexp 455	. . . . . . . . . . . . . . . . . 18 ⊢ (((([,]‘𝑡) ⊆ ([,]‘𝑤) ∧ ([,]‘𝑚) ⊆ ([,]‘𝑤)) → 𝑚 = 𝑤) ↔ (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
45		sstr2 3944	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (([,]‘𝑡) ⊆ ([,]‘𝑚) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
46	45	ad2antll 741	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
47	46	ancrd 560	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → (([,]‘𝑡) ⊆ ([,]‘𝑤) ∧ ([,]‘𝑚) ⊆ ([,]‘𝑤))))
48	47	imim1d 83	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (((([,]‘𝑡) ⊆ ([,]‘𝑤) ∧ ([,]‘𝑚) ⊆ ([,]‘𝑤)) → 𝑚 = 𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
49	44, 48	biimtrrid 246	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → ((([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
50	49	imim2d 58	. . . . . . . . . . . . . . . 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	biimtrid 245	. . . . . . . . . . . . . . 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 3174	. . . . . . . . . . . . . 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 459	. . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑚 → ([,]‘𝑧) = ([,]‘𝑚))
55	54	sseq1d 3968	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑚 → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘𝑚) ⊆ ([,]‘𝑤)))
56		equequ1 2055	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑚 → (𝑧 = 𝑤 ↔ 𝑚 = 𝑤))
57	55, 56	imbi12d 347	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑚 → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
58	57	ralbidv 3188	. . . . . . . . . . . . . 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 3654	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝐺 ↔ (𝑚 ∈ 𝐴 ∧ ∀𝑤 ∈ 𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
61	37, 53, 60	sylanbrc 594	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚 ∈ 𝐺)
62		ffun 6708	. . . . . . . . . . . . . 14 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → Fun [,])
63	2, 62	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun [,]
64	59	ssrab3 4036	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 ⊆ 𝐴
65	64, 14	sstrid 3948	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ⊆ (ℝ^* × ℝ^*))
66	2	fdmi 6717	. . . . . . . . . . . . . . 15 ⊢ dom [,] = (ℝ^* × ℝ^*)
67	65, 66	sseqtrrdi 3978	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ⊆ dom [,])
68	67	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝐺 ⊆ dom [,])
69		funfvima2 7229	. . . . . . . . . . . . 13 ⊢ ((Fun [,] ∧ 𝐺 ⊆ dom [,]) → (𝑚 ∈ 𝐺 → ([,]‘𝑚) ∈ ([,] “ 𝐺)))
70	63, 68, 69	sylancr 598	. . . . . . . . . . . 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 16	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ([,]‘𝑚) ∈ ([,] “ 𝐺))
72		elunii 4877	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ([,]‘𝑚) ∧ ([,]‘𝑚) ∈ ([,] “ 𝐺)) → 𝑎 ∈ ∪ ([,] “ 𝐺))
73	36, 71, 72	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ∪ ([,] “ 𝐺))
74	73	exp32 425	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) → (∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ∈ ∪ ([,] “ 𝐺))))
75	33, 74	biimtrid 245	. . . . . . . 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 3164	. . . . . . 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 16	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → 𝑎 ∈ ∪ ([,] “ 𝐺))
78	77	rexlimdvaa 3167	. . . . 5 ⊢ (𝜑 → (∃𝑡 ∈ 𝐴 𝑎 ∈ ([,]‘𝑡) → 𝑎 ∈ ∪ ([,] “ 𝐺)))
79	17, 78	sylbid 243	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖 → 𝑎 ∈ ∪ ([,] “ 𝐺)))
80	1, 79	biimtrid 245	. . 3 ⊢ (𝜑 → (𝑎 ∈ ∪ ([,] “ 𝐴) → 𝑎 ∈ ∪ ([,] “ 𝐺)))
81	80	ssrdv 3943	. 2 ⊢ (𝜑 → ∪ ([,] “ 𝐴) ⊆ ∪ ([,] “ 𝐺))
82		imass2 6104	. . . 4 ⊢ (𝐺 ⊆ 𝐴 → ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
83	64, 82	ax-mp 5	. . 3 ⊢ ([,] “ 𝐺) ⊆ ([,] “ 𝐴)
84		uniss 4880	. . 3 ⊢ (([,] “ 𝐺) ⊆ ([,] “ 𝐴) → ∪ ([,] “ 𝐺) ⊆ ∪ ([,] “ 𝐴))
85	83, 84	mp1i 14	. 2 ⊢ (𝜑 → ∪ ([,] “ 𝐺) ⊆ ∪ ([,] “ 𝐴))
86	81, 85	eqssd 3954	1 ⊢ (𝜑 → ∪ ([,] “ 𝐴) = ∪ ([,] “ 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 {crab 3416 ∩ cin 3904 ⊆ wss 3905 ∅c0 4286 𝒫 cpw 4562 ⟨cop 4595 ∪ cuni 4872 × cxp 5659 dom cdm 5661 ran crn 5662 “ cima 5664 Fun wfun 6530 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ∈ cmpo 7412 ℝcr 11094 1c1 11096 + caddc 11098 ℝ^*cxr 11237 ≤ cle 11239 / cdiv 11866 2c2 12290 ℕ₀cn0 12499 ℤcz 12586 [,]cicc 13370 ↑cexp 14093

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-n0 12500 df-z 12587 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-clim 15535 df-sum 15734 df-rest 17470 df-topgen 17491 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-top 23051 df-topon 23068 df-bases 23103 df-cmp 23544 df-ovol 25623

This theorem is referenced by: dyadmbl 25759 mblfinlem1 38308 mblfinlem2 38309

W3C validator