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Theorem dyadmbllem 24468

Description: Lemma for dyadmbl 24469. (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 4813	. . . 4 ⊢ (𝑎 ∈ ∪ ([,] “ 𝐴) ↔ ∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖)
2		iccf 13019	. . . . . . 7 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
3		ffn 6534	. . . . . . 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 24460	. . . . . . . . 9 ⊢ 𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ))
8		frn 6541	. . . . . . . . 9 ⊢ (𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
9	7, 8	ax-mp 5	. . . . . . . 8 ⊢ ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
10		inss2 4134	. . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
11		rexpssxrxp 10861	. . . . . . . . 9 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
12	10, 11	sstri 3900	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
13	9, 12	sstri 3900	. . . . . . 7 ⊢ ran 𝐹 ⊆ (ℝ^* × ℝ^*)
14	5, 13	sstrdi 3903	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ (ℝ^* × ℝ^*))
15		eleq2 2822	. . . . . . 7 ⊢ (𝑖 = ([,]‘𝑡) → (𝑎 ∈ 𝑖 ↔ 𝑎 ∈ ([,]‘𝑡)))
16	15	rexima 7042	. . . . . 6 ⊢ (([,] Fn (ℝ^* × ℝ^) ∧ 𝐴 ⊆ (ℝ^ × ℝ^*)) → (∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖 ↔ ∃𝑡 ∈ 𝐴 𝑎 ∈ ([,]‘𝑡)))
17	4, 14, 16	sylancr 590	. . . . 5 ⊢ (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖 ↔ ∃𝑡 ∈ 𝐴 𝑎 ∈ ([,]‘𝑡)))
18		ssrab2 3983	. . . . . . . . 9 ⊢ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ 𝐴
19	5	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → 𝐴 ⊆ ran 𝐹)
20	18, 19	sstrid 3902	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ⊆ ran 𝐹)
21		simprl 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → 𝑡 ∈ 𝐴)
22		ssid 3913	. . . . . . . . . 10 ⊢ ([,]‘𝑡) ⊆ ([,]‘𝑡)
23		fveq2 6706	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑡 → ([,]‘𝑎) = ([,]‘𝑡))
24	23	sseq2d 3923	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑡 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑡)))
25	24	rspcev 3530	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑡)) → ∃𝑎 ∈ 𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
26	21, 22, 25	sylancl 589	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → ∃𝑎 ∈ 𝐴 ([,]‘𝑡) ⊆ ([,]‘𝑎))
27		rabn0 4290	. . . . . . . . 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 24467	. . . . . . . 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 587	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) → ∃𝑚 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)}∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))
31		fveq2 6706	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑚 → ([,]‘𝑎) = ([,]‘𝑚))
32	31	sseq2d 3923	. . . . . . . . . 10 ⊢ (𝑎 = 𝑚 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
33	32	elrab 3595	. . . . . . . . 9 ⊢ (𝑚 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)))
34		simprlr 780	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → ([,]‘𝑡) ⊆ ([,]‘𝑚))
35		simplrr 778	. . . . . . . . . . . 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 3892	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑎 ∈ ([,]‘𝑚))
37		simprll 779	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚 ∈ 𝐴)
38		fveq2 6706	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑤 → ([,]‘𝑎) = ([,]‘𝑤))
39	38	sseq2d 3923	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑤 → (([,]‘𝑡) ⊆ ([,]‘𝑎) ↔ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
40	39	elrab 3595	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} ↔ (𝑤 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑤)))
41	40	imbi1i 353	. . . . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . . . 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 3898	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (([,]‘𝑡) ⊆ ([,]‘𝑚) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
46	45	ad2antll 729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ (𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚))) → (([,]‘𝑚) ⊆ ([,]‘𝑤) → ([,]‘𝑡) ⊆ ([,]‘𝑤)))
47	46	ancrd 555	. . . . . . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . . . . 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 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 3092	. . . . . . . . . . . . . 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 6706	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑚 → ([,]‘𝑧) = ([,]‘𝑚))
55	54	sseq1d 3922	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑚 → (([,]‘𝑧) ⊆ ([,]‘𝑤) ↔ ([,]‘𝑚) ⊆ ([,]‘𝑤)))
56		equequ1 2033	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑚 → (𝑧 = 𝑤 ↔ 𝑚 = 𝑤))
57	55, 56	imbi12d 348	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑚 → ((([,]‘𝑧) ⊆ ([,]‘𝑤) → 𝑧 = 𝑤) ↔ (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
58	57	ralbidv 3111	. . . . . . . . . . . . . 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 3598	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝐺 ↔ (𝑚 ∈ 𝐴 ∧ ∀𝑤 ∈ 𝐴 (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤)))
61	37, 53, 60	sylanbrc 586	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝑚 ∈ 𝐺)
62		ffun 6537	. . . . . . . . . . . . . 14 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → Fun [,])
63	2, 62	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun [,]
64	59	ssrab3 3985	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 ⊆ 𝐴
65	64, 14	sstrid 3902	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ⊆ (ℝ^* × ℝ^*))
66	2	fdmi 6546	. . . . . . . . . . . . . . 15 ⊢ dom [,] = (ℝ^* × ℝ^*)
67	65, 66	sseqtrrdi 3942	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ⊆ dom [,])
68	67	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑡 ∈ 𝐴 ∧ 𝑎 ∈ ([,]‘𝑡))) ∧ ((𝑚 ∈ 𝐴 ∧ ([,]‘𝑡) ⊆ ([,]‘𝑚)) ∧ ∀𝑤 ∈ {𝑎 ∈ 𝐴 ∣ ([,]‘𝑡) ⊆ ([,]‘𝑎)} (([,]‘𝑚) ⊆ ([,]‘𝑤) → 𝑚 = 𝑤))) → 𝐺 ⊆ dom [,])
69		funfvima2 7036	. . . . . . . . . . . . 13 ⊢ ((Fun [,] ∧ 𝐺 ⊆ dom [,]) → (𝑚 ∈ 𝐺 → ([,]‘𝑚) ∈ ([,] “ 𝐺)))
70	63, 68, 69	sylancr 590	. . . . . . . . . . . 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 4814	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ([,]‘𝑚) ∧ ([,]‘𝑚) ∈ ([,] “ 𝐺)) → 𝑎 ∈ ∪ ([,] “ 𝐺))
73	36, 71, 72	syl2anc 587	. . . . . . . . . 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	syl5bi 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 3195	. . . . . . 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 3197	. . . . 5 ⊢ (𝜑 → (∃𝑡 ∈ 𝐴 𝑎 ∈ ([,]‘𝑡) → 𝑎 ∈ ∪ ([,] “ 𝐺)))
79	17, 78	sylbid 243	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ ([,] “ 𝐴)𝑎 ∈ 𝑖 → 𝑎 ∈ ∪ ([,] “ 𝐺)))
80	1, 79	syl5bi 245	. . 3 ⊢ (𝜑 → (𝑎 ∈ ∪ ([,] “ 𝐴) → 𝑎 ∈ ∪ ([,] “ 𝐺)))
81	80	ssrdv 3897	. 2 ⊢ (𝜑 → ∪ ([,] “ 𝐴) ⊆ ∪ ([,] “ 𝐺))
82		imass2 5959	. . . 4 ⊢ (𝐺 ⊆ 𝐴 → ([,] “ 𝐺) ⊆ ([,] “ 𝐴))
83	64, 82	ax-mp 5	. . 3 ⊢ ([,] “ 𝐺) ⊆ ([,] “ 𝐴)
84		uniss 4817	. . 3 ⊢ (([,] “ 𝐺) ⊆ ([,] “ 𝐴) → ∪ ([,] “ 𝐺) ⊆ ∪ ([,] “ 𝐴))
85	83, 84	mp1i 13	. 2 ⊢ (𝜑 → ∪ ([,] “ 𝐺) ⊆ ∪ ([,] “ 𝐴))
86	81, 85	eqssd 3908	1 ⊢ (𝜑 → ∪ ([,] “ 𝐴) = ∪ ([,] “ 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ∀wral 3054 ∃wrex 3055 {crab 3058 ∩ cin 3856 ⊆ wss 3857 ∅c0 4227 𝒫 cpw 4503 ⟨cop 4537 ∪ cuni 4809 × cxp 5538 dom cdm 5540 ran crn 5541 “ cima 5543 Fun wfun 6363 Fn wfn 6364 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ∈ cmpo 7204 ℝcr 10711 1c1 10713 + caddc 10715 ℝ^*cxr 10849 ≤ cle 10851 / cdiv 11472 2c2 11868 ℕ₀cn0 12073 ℤcz 12159 [,]cicc 12921 ↑cexp 13618

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fi 9016 df-sup 9047 df-inf 9048 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-z 12160 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ioo 12922 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-clim 15032 df-sum 15233 df-rest 16899 df-topgen 16920 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-top 21763 df-topon 21780 df-bases 21815 df-cmp 22256 df-ovol 24333

This theorem is referenced by: dyadmbl 24469 mblfinlem1 35508 mblfinlem2 35509

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