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Theorem dyadmbllem 25649

Description: Lemma for dyadmbl 25650. (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 4866	. . . 4 ⊢ (𝑎 ∈ ∪ ([,] “ 𝐴) ↔ ∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖)
2		iccf 13446	. . . . . . 7 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
3		ffn 6686	. . . . . . 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 25641	. . . . . . . . 9 ⊢ 𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ))
8		frn 6694	. . . . . . . . 9 ⊢ (𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
9	7, 8	ax-mp 5	. . . . . . . 8 ⊢ ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
10		inss2 4187	. . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
11		rexpssxrxp 11221	. . . . . . . . 9 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
12	10, 11	sstri 3943	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
13	9, 12	sstri 3943	. . . . . . 7 ⊢ ran 𝐹 ⊆ (ℝ^* × ℝ^*)
14	5, 13	sstrdi 3946	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (ℝ^* × ℝ^*))
15		eleq2 2850	. . . . . . 7 ⊢ (𝑖 = ([,]‘𝑡) → (𝑎 ∈ 𝑖 ↔ 𝑎 ∈ ([,]‘𝑡)))
16	15	rexima 7217	. . . . . 6 ⊢ (([,] Fn (ℝ^* × ℝ^) ∧ 𝐴 ⊆ (ℝ^ × ℝ^*)) → (∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖 ↔ ∃𝑡 ∈ 𝐴 𝑎 ∈ ([,]‘𝑡)))
17	4, 14, 16	sylancr 596	. . . . 5 ⊢ (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖 ↔ ∃𝑡 ∈ 𝐴 𝑎 ∈ ([,]‘𝑡)))
18		ssrab2 4031	. . . . . . . . 9 ⊢ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ 𝐴
19	5	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → 𝐴 ⊆ ran 𝐹)
20	18, 19	sstrid 3945	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ ran 𝐹)
21		simprl 780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → 𝑡 ∈ 𝐴)
22		ssid 3956	. . . . . . . . . 10 ⊢ ([,]‘𝑡) ⊆ ([,]‘𝑡)
23		fveq2 6862	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑡 → ([,]‘𝑎) = ([,]‘𝑡))
24	23	sseq2d 3966	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑡 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑡)))
25	24	rspcev 3580	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑡)) → ∃𝑎 ∈ 𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
26	21, 22, 25	sylancl 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → ∃𝑎 ∈ 𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
27		rabn0 4340	. . . . . . . . 9 ⊢ ({𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅ ↔ ∃𝑎 ∈ 𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
28	26, 27	sylibr 236	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ≠ ∅)
29	6	dyadmax 25648	. . . . . . . 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 593	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → ∃𝑚 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
31		fveq2 6862	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑚 → ([,]‘𝑎) = ([,]‘𝑚))
32	31	sseq2d 3966	. . . . . . . . . 10 ⊢ (𝑎 = 𝑚 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
33	32	elrab 3649	. . . . . . . . 9 ⊢ (𝑚 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
34		simprlr 789	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ([,]‘𝑡) ⊆ ([,]‘𝑚))
35		simplrr 787	. . . . . . . . . . . 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 3935	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ([,]‘𝑚))
37		simprll 788	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚 ∈ 𝐴)
38		fveq2 6862	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑤 → ([,]‘𝑎) = ([,]‘𝑤))
39	38	sseq2d 3966	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑤 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
40	39	elrab 3649	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑤 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
41	40	imbi1i 351	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ ((𝑤 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
42		impexp 454	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ (𝑤 ∈ 𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
43	41, 42	bitri 277	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)) ↔ (𝑤 ∈ 𝐴 → (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))))
44		impexp 454	. . . . . . . . . . . . . . . . . 18 ⊢ (((([,]‘𝑡) ⊆ ([,]‘𝑤) ∧ ([,]‘𝑚) ⊆ ([,]‘𝑤)) → 𝑚 = 𝑤) ↔ (([,]‘𝑡) ⊆ ([,]‘𝑤) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
45		sstr2 3941	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (([,]‘𝑡) ⊆ ([,]‘𝑚) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
46	45	ad2antll 739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
47	46	ancrd 559	. . . . . . . . . . . . . . . . . . 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	biimtrrid 245	. . . . . . . . . . . . . . . . 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	biimtrid 244	. . . . . . . . . . . . . . 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 3170	. . . . . . . . . . . . . 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 458	. . . . . . . . . . . . 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 6862	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑚 → ([,]‘𝑧) = ([,]‘𝑚))
55	54	sseq1d 3965	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑚 → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘𝑚) ⊆ ([,]‘𝑤)))
56		equequ1 2044	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑚 → (𝑧 = 𝑤 ↔ 𝑚 = 𝑤))
57	55, 56	imbi12d 346	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑚 → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
58	57	ralbidv 3184	. . . . . . . . . . . . . 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 3652	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝐺 ↔ (𝑚 ∈ 𝐴 ∧ ∀𝑤 ∈ 𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
61	37, 53, 60	sylanbrc 592	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚 ∈ 𝐺)
62		ffun 6689	. . . . . . . . . . . . . 14 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → Fun [,])
63	2, 62	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun [,]
64	59	ssrab3 4033	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 ⊆ 𝐴
65	64, 14	sstrid 3945	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ⊆ (ℝ^* × ℝ^*))
66	2	fdmi 6698	. . . . . . . . . . . . . . 15 ⊢ dom [,] = (ℝ^* × ℝ^*)
67	65, 66	sseqtrrdi 3975	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ⊆ dom [,])
68	67	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝐺 ⊆ dom [,])
69		funfvima2 7210	. . . . . . . . . . . . 13 ⊢ ((Fun [,] ∧ 𝐺 ⊆ dom [,]) → (𝑚 ∈ 𝐺 → ([,]‘𝑚) ∈ ([,] “ 𝐺)))
70	63, 68, 69	sylancr 596	. . . . . . . . . . . 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 4867	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ([,]‘𝑚) ∧ ([,]‘𝑚) ∈ ([,] “ 𝐺)) → 𝑎 ∈ ∪ ([,] “ 𝐺))
73	36, 71, 72	syl2anc 593	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ∪ ([,] “ 𝐺))
74	73	exp32 424	. . . . . . . . 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 244	. . . . . . . 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 3160	. . . . . . 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 3163	. . . . 5 ⊢ (𝜑 → (∃𝑡 ∈ 𝐴 𝑎 ∈ ([,]‘𝑡) → 𝑎 ∈ ∪ ([,] “ 𝐺)))
79	17, 78	sylbid 242	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖 → 𝑎 ∈ ∪ ([,] “ 𝐺)))
80	1, 79	biimtrid 244	. . 3 ⊢ (𝜑 → (𝑎 ∈ ∪ ([,] “ 𝐴) → 𝑎 ∈ ∪ ([,] “ 𝐺)))
81	80	ssrdv 3940	. 2 ⊢ (𝜑 → ∪ ([,] “ 𝐴) ⊆ ∪ ([,] “ 𝐺))
82		imass2 6087	. . . 4 ⊢ (𝐺 ⊆ 𝐴 → ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
83	64, 82	ax-mp 5	. . 3 ⊢ ([,] “ 𝐺) ⊆ ([,] “ 𝐴)
84		uniss 4870	. . 3 ⊢ (([,] “ 𝐺) ⊆ ([,] “ 𝐴) → ∪ ([,] “ 𝐺) ⊆ ∪ ([,] “ 𝐴))
85	83, 84	mp1i 13	. 2 ⊢ (𝜑 → ∪ ([,] “ 𝐺) ⊆ ∪ ([,] “ 𝐴))
86	81, 85	eqssd 3951	1 ⊢ (𝜑 → ∪ ([,] “ 𝐴) = ∪ ([,] “ 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 ∃wrex 3085 {crab 3413 ∩ cin 3901 ⊆ wss 3902 ∅c0 4283 𝒫 cpw 4552 ⟨cop 4585 ∪ cuni 4862 × cxp 5641 dom cdm 5643 ran crn 5644 “ cima 5646 Fun wfun 6510 Fn wfn 6511 ⟶wf 6512 ‘cfv 6516 (class class class)co 7391 ∈ cmpo 7393 ℝcr 11066 1c1 11068 + caddc 11070 ℝ^*cxr 11209 ≤ cle 11211 / cdiv 11838 2c2 12266 ℕ₀cn0 12475 ℤcz 12562 [,]cicc 13346 ↑cexp 14068

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-inf2 9590 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-er 8672 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fi 9351 df-sup 9382 df-inf 9383 df-oi 9452 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-n0 12476 df-z 12563 df-uz 12834 df-q 12944 df-rp 12988 df-xneg 13108 df-xadd 13109 df-xmul 13110 df-ioo 13347 df-ico 13349 df-icc 13350 df-fz 13507 df-fzo 13654 df-seq 14009 df-exp 14069 df-hash 14338 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254 df-clim 15506 df-sum 15705 df-rest 17442 df-topgen 17463 df-psmet 21404 df-xmet 21405 df-met 21406 df-bl 21407 df-mopn 21408 df-top 22942 df-topon 22959 df-bases 22994 df-cmp 23435 df-ovol 25514

This theorem is referenced by: dyadmbl 25650 mblfinlem1 38117 mblfinlem2 38118

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