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Theorem dyadmbllem 23173

Description: Lemma for dyadmbl 23174. (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 4376	. . . 4 ⊢ (𝑎 ∈ ∪ ([,] “ 𝐴) ↔ ∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖)
2		iccf 12143	. . . . . . 7 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
3		ffn 5958	. . . . . . 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 23165	. . . . . . . . 9 ⊢ 𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ))
8		frn 5966	. . . . . . . . 9 ⊢ (𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
9	7, 8	ax-mp 5	. . . . . . . 8 ⊢ ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
10		inss2 3796	. . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
11		rexpssxrxp 9963	. . . . . . . . 9 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
12	10, 11	sstri 3577	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
13	9, 12	sstri 3577	. . . . . . 7 ⊢ ran 𝐹 ⊆ (ℝ^* × ℝ^*)
14	5, 13	syl6ss 3580	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (ℝ^* × ℝ^*))
15		eleq2 2677	. . . . . . 7 ⊢ (𝑖 = ([,]‘𝑡) → (𝑎 ∈ 𝑖 ↔ 𝑎 ∈ ([,]‘𝑡)))
16	15	rexima 6401	. . . . . 6 ⊢ (([,] Fn (ℝ^* × ℝ^) ∧ 𝐴 ⊆ (ℝ^ × ℝ^*)) → (∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖 ↔ ∃𝑡 ∈ 𝐴 𝑎 ∈ ([,]‘𝑡)))
17	4, 14, 16	sylancr 694	. . . . 5 ⊢ (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖 ↔ ∃𝑡 ∈ 𝐴 𝑎 ∈ ([,]‘𝑡)))
18		ssrab2 3650	. . . . . . . . 9 ⊢ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ 𝐴
19	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → 𝐴 ⊆ ran 𝐹)
20	18, 19	syl5ss 3579	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ ran 𝐹)
21		simprl 790	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → 𝑡 ∈ 𝐴)
22		ssid 3587	. . . . . . . . . 10 ⊢ ([,]‘𝑡) ⊆ ([,]‘𝑡)
23		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑡 → ([,]‘𝑎) = ([,]‘𝑡))
24	23	sseq2d 3596	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑡 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑡)))
25	24	rspcev 3282	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑡)) → ∃𝑎 ∈ 𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
26	21, 22, 25	sylancl 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → ∃𝑎 ∈ 𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
27		rabn0 3912	. . . . . . . . 9 ⊢ ({𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅ ↔ ∃𝑎 ∈ 𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
28	26, 27	sylibr 223	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅)
29	6	dyadmax 23172	. . . . . . . 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 691	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → ∃𝑚 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
31		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑚 → ([,]‘𝑎) = ([,]‘𝑚))
32	31	sseq2d 3596	. . . . . . . . . 10 ⊢ (𝑎 = 𝑚 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
33	32	elrab 3331	. . . . . . . . 9 ⊢ (𝑚 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
34		simprlr 799	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ([,]‘𝑡) ⊆ ([,]‘𝑚))
35		simplrr 797	. . . . . . . . . . . 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 3569	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ([,]‘𝑚))
37		simprll 798	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚 ∈ 𝐴)
38		fveq2 6103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑤 → ([,]‘𝑎) = ([,]‘𝑤))
39	38	sseq2d 3596	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑤 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
40	39	elrab 3331	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑤 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
41	40	imbi1i 338	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ ((𝑤 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
42		impexp 461	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ (𝑤 ∈ 𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
43	41, 42	bitri 263	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ (𝑤 ∈ 𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
44		impexp 461	. . . . . . . . . . . . . . . . . 18 ⊢ (((([,]‘𝑡) ⊆ ([,]‘𝑤) ∧ ([,]‘𝑚) ⊆ ([,]‘𝑤)) → 𝑚 = 𝑤) ↔ (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
45		sstr2 3575	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (([,]‘𝑡) ⊆ ([,]‘𝑚) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
46	45	ad2antll 761	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
47	46	ancrd 575	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → (([,]‘𝑡) ⊆ ([,]‘𝑤) ∧ ([,]‘𝑚) ⊆ ([,]‘𝑤))))
48	47	imim1d 80	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (((([,]‘𝑡) ⊆ ([,]‘𝑤) ∧ ([,]‘𝑚) ⊆ ([,]‘𝑤)) → 𝑚 = 𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
49	44, 48	syl5bir 232	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → ((([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
50	49	imim2d 55	. . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . 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 2944	. . . . . . . . . . . . . 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 647	. . . . . . . . . . . . 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 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑚 → ([,]‘𝑧) = ([,]‘𝑚))
55	54	sseq1d 3595	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑚 → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘𝑚) ⊆ ([,]‘𝑤)))
56		equequ1 1939	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑚 → (𝑧 = 𝑤 ↔ 𝑚 = 𝑤))
57	55, 56	imbi12d 333	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑚 → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
58	57	ralbidv 2969	. . . . . . . . . . . . . 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 3333	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝐺 ↔ (𝑚 ∈ 𝐴 ∧ ∀𝑤 ∈ 𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
61	37, 53, 60	sylanbrc 695	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚 ∈ 𝐺)
62		ffun 5961	. . . . . . . . . . . . . 14 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → Fun [,])
63	2, 62	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun [,]
64		ssrab2 3650	. . . . . . . . . . . . . . . . 17 ⊢ {𝑧 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐴 (([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤)} ⊆ 𝐴
65	59, 64	eqsstri 3598	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 ⊆ 𝐴
66	65, 14	syl5ss 3579	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ⊆ (ℝ^* × ℝ^*))
67	2	fdmi 5965	. . . . . . . . . . . . . . 15 ⊢ dom [,] = (ℝ^* × ℝ^*)
68	66, 67	syl6sseqr 3615	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ⊆ dom [,])
69	68	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝐺 ⊆ dom [,])
70		funfvima2 6397	. . . . . . . . . . . . 13 ⊢ ((Fun [,] ∧ 𝐺 ⊆ dom [,]) → (𝑚 ∈ 𝐺 → ([,]‘𝑚) ∈ ([,] “ 𝐺)))
71	63, 69, 70	sylancr 694	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → (𝑚 ∈ 𝐺 → ([,]‘𝑚) ∈ ([,] “ 𝐺)))
72	61, 71	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ([,]‘𝑚) ∈ ([,] “ 𝐺))
73		elunii 4377	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ([,]‘𝑚) ∧ ([,]‘𝑚) ∈ ([,] “ 𝐺)) → 𝑎 ∈ ∪ ([,] “ 𝐺))
74	36, 72, 73	syl2anc 691	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ∪ ([,] “ 𝐺))
75	74	exp32 629	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) → (∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ∈ ∪ ([,] “ 𝐺))))
76	33, 75	syl5bi 231	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → (𝑚 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ∈ ∪ ([,] “ 𝐺))))
77	76	rexlimdv 3012	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → (∃𝑚 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤) → 𝑎 ∈ ∪ ([,] “ 𝐺)))
78	30, 77	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → 𝑎 ∈ ∪ ([,] “ 𝐺))
79	78	rexlimdvaa 3014	. . . . 5 ⊢ (𝜑 → (∃𝑡 ∈ 𝐴 𝑎 ∈ ([,]‘𝑡) → 𝑎 ∈ ∪ ([,] “ 𝐺)))
80	17, 79	sylbid 229	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖 → 𝑎 ∈ ∪ ([,] “ 𝐺)))
81	1, 80	syl5bi 231	. . 3 ⊢ (𝜑 → (𝑎 ∈ ∪ ([,] “ 𝐴) → 𝑎 ∈ ∪ ([,] “ 𝐺)))
82	81	ssrdv 3574	. 2 ⊢ (𝜑 → ∪ ([,] “ 𝐴) ⊆ ∪ ([,] “ 𝐺))
83		imass2 5420	. . . 4 ⊢ (𝐺 ⊆ 𝐴 → ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
84	65, 83	ax-mp 5	. . 3 ⊢ ([,] “ 𝐺) ⊆ ([,] “ 𝐴)
85		uniss 4394	. . 3 ⊢ (([,] “ 𝐺) ⊆ ([,] “ 𝐴) → ∪ ([,] “ 𝐺) ⊆ ∪ ([,] “ 𝐴))
86	84, 85	mp1i 13	. 2 ⊢ (𝜑 → ∪ ([,] “ 𝐺) ⊆ ∪ ([,] “ 𝐴))
87	82, 86	eqssd 3585	1 ⊢ (𝜑 → ∪ ([,] “ 𝐴) = ∪ ([,] “ 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 ⟨cop 4131 ∪ cuni 4372 × cxp 5036 dom cdm 5038 ran crn 5039 “ cima 5041 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ℝcr 9814 1c1 9816 + caddc 9818 ℝ^*cxr 9952 ≤ cle 9954 / cdiv 10563 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 [,]cicc 12049 ↑cexp 12722

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-rest 15906 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cmp 21000 df-ovol 23040

This theorem is referenced by: dyadmbl 23174 mblfinlem1 32616 mblfinlem2 32617

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