cvmlift2lem13 - Mathbox for Mario Carneiro

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Theorem cvmlift2lem13 35629

Description: Lemma for cvmlift2 35630. (Contributed by Mario Carneiro, 7-May-2015.)

**Hypotheses**
Ref	Expression
cvmlift2.b	⊢ 𝐵 = ∪ 𝐶
cvmlift2.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmlift2.g	⊢ (𝜑 → 𝐺 ∈ ((II ×_t II) Cn 𝐽))
cvmlift2.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
cvmlift2.i	⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))
cvmlift2.h	⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))
cvmlift2.k	⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))

**Assertion**
Ref	Expression
cvmlift2lem13	⊢ (𝜑 → ∃!𝑔 ∈ ((II ×_t II) Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (0𝑔0) = 𝑃))

Distinct variable groups: 𝑓,𝑔,𝑥,𝑦,𝑧,𝐹 𝜑,𝑓,𝑔,𝑥,𝑦,𝑧 𝑓,𝐽,𝑔,𝑥,𝑦,𝑧 𝑓,𝐺,𝑔,𝑥,𝑦,𝑧 𝑓,𝐻,𝑥,𝑦,𝑧 𝐶,𝑓,𝑔,𝑥,𝑦,𝑧 𝑃,𝑓,𝑔,𝑥,𝑦,𝑧 𝑥,𝐵,𝑦,𝑧 𝑓,𝐾,𝑔,𝑥,𝑦,𝑧 Allowed substitution hints: 𝐵(𝑓,𝑔) 𝐻(𝑔)

Proof of Theorem cvmlift2lem13 Dummy variables 𝑏 𝑐 𝑑 𝑢 𝑣 𝑎 𝑟 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmlift2.b	. . . 4 ⊢ 𝐵 = ∪ 𝐶
2		cvmlift2.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
3		cvmlift2.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ ((II ×_t II) Cn 𝐽))
4		cvmlift2.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
5		cvmlift2.i	. . . 4 ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))
6		cvmlift2.h	. . . 4 ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))
7		cvmlift2.k	. . . 4 ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))
8		fveq2 6863	. . . . . 6 ⊢ (𝑎 = 𝑧 → (((II ×_t II) CnP 𝐶)‘𝑎) = (((II ×_t II) CnP 𝐶)‘𝑧))
9	8	eleq2d 2847	. . . . 5 ⊢ (𝑎 = 𝑧 → (𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎) ↔ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧)))
10	9	cbvrabv 3423	. . . 4 ⊢ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} = {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑧)}
11		sneq 4591	. . . . . . 7 ⊢ (𝑧 = 𝑏 → {𝑧} = {𝑏})
12	11	xpeq2d 5675	. . . . . 6 ⊢ (𝑧 = 𝑏 → ((0[,]1) × {𝑧}) = ((0[,]1) × {𝑏}))
13	12	sseq1d 3967	. . . . 5 ⊢ (𝑧 = 𝑏 → (((0[,]1) × {𝑧}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ ((0[,]1) × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)}))
14	13	cbvrabv 3423	. . . 4 ⊢ {𝑧 ∈ (0[,]1) ∣ ((0[,]1) × {𝑧}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)}} = {𝑏 ∈ (0[,]1) ∣ ((0[,]1) × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)}}
15		simpr 488	. . . . . . 7 ⊢ ((𝑐 = 𝑟 ∧ 𝑑 = 𝑡) → 𝑑 = 𝑡)
16	15	eleq1d 2846	. . . . . 6 ⊢ ((𝑐 = 𝑟 ∧ 𝑑 = 𝑡) → (𝑑 ∈ (0[,]1) ↔ 𝑡 ∈ (0[,]1)))
17		xpeq1 5659	. . . . . . . . . 10 ⊢ (𝑣 = 𝑢 → (𝑣 × {𝑏}) = (𝑢 × {𝑏}))
18	17	sseq1d 3967	. . . . . . . . 9 ⊢ (𝑣 = 𝑢 → ((𝑣 × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ (𝑢 × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)}))
19		xpeq1 5659	. . . . . . . . . 10 ⊢ (𝑣 = 𝑢 → (𝑣 × {𝑑}) = (𝑢 × {𝑑}))
20	19	sseq1d 3967	. . . . . . . . 9 ⊢ (𝑣 = 𝑢 → ((𝑣 × {𝑑}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ (𝑢 × {𝑑}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)}))
21	18, 20	bibi12d 347	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (((𝑣 × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ (𝑣 × {𝑑}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)}) ↔ ((𝑢 × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ (𝑢 × {𝑑}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)})))
22	21	cbvrexvw 3240	. . . . . . 7 ⊢ (∃𝑣 ∈ ((nei‘II)‘{𝑐})((𝑣 × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ (𝑣 × {𝑑}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)}) ↔ ∃𝑢 ∈ ((nei‘II)‘{𝑐})((𝑢 × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ (𝑢 × {𝑑}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)}))
23		simpl 486	. . . . . . . . . 10 ⊢ ((𝑐 = 𝑟 ∧ 𝑑 = 𝑡) → 𝑐 = 𝑟)
24	23	sneqd 4593	. . . . . . . . 9 ⊢ ((𝑐 = 𝑟 ∧ 𝑑 = 𝑡) → {𝑐} = {𝑟})
25	24	fveq2d 6867	. . . . . . . 8 ⊢ ((𝑐 = 𝑟 ∧ 𝑑 = 𝑡) → ((nei‘II)‘{𝑐}) = ((nei‘II)‘{𝑟}))
26	15	sneqd 4593	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝑟 ∧ 𝑑 = 𝑡) → {𝑑} = {𝑡})
27	26	xpeq2d 5675	. . . . . . . . . 10 ⊢ ((𝑐 = 𝑟 ∧ 𝑑 = 𝑡) → (𝑢 × {𝑑}) = (𝑢 × {𝑡}))
28	27	sseq1d 3967	. . . . . . . . 9 ⊢ ((𝑐 = 𝑟 ∧ 𝑑 = 𝑡) → ((𝑢 × {𝑑}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ (𝑢 × {𝑡}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)}))
29	28	bibi2d 344	. . . . . . . 8 ⊢ ((𝑐 = 𝑟 ∧ 𝑑 = 𝑡) → (((𝑢 × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ (𝑢 × {𝑑}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)}) ↔ ((𝑢 × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ (𝑢 × {𝑡}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)})))
30	25, 29	rexeqbidv 3336	. . . . . . 7 ⊢ ((𝑐 = 𝑟 ∧ 𝑑 = 𝑡) → (∃𝑢 ∈ ((nei‘II)‘{𝑐})((𝑢 × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ (𝑢 × {𝑑}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)}) ↔ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ (𝑢 × {𝑡}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)})))
31	22, 30	bitrid 285	. . . . . 6 ⊢ ((𝑐 = 𝑟 ∧ 𝑑 = 𝑡) → (∃𝑣 ∈ ((nei‘II)‘{𝑐})((𝑣 × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ (𝑣 × {𝑑}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)}) ↔ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ (𝑢 × {𝑡}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)})))
32	16, 31	anbi12d 641	. . . . 5 ⊢ ((𝑐 = 𝑟 ∧ 𝑑 = 𝑡) → ((𝑑 ∈ (0[,]1) ∧ ∃𝑣 ∈ ((nei‘II)‘{𝑐})((𝑣 × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ (𝑣 × {𝑑}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)})) ↔ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ (𝑢 × {𝑡}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)}))))
33	32	cbvopabv 5172	. . . 4 ⊢ {⟨𝑐, 𝑑⟩ ∣ (𝑑 ∈ (0[,]1) ∧ ∃𝑣 ∈ ((nei‘II)‘{𝑐})((𝑣 × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ (𝑣 × {𝑑}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)}))} = {⟨𝑟, 𝑡⟩ ∣ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑏}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)} ↔ (𝑢 × {𝑡}) ⊆ {𝑎 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×_t II) CnP 𝐶)‘𝑎)}))}
34	1, 2, 3, 4, 5, 6, 7, 10, 14, 33	cvmlift2lem12 35628	. . 3 ⊢ (𝜑 → 𝐾 ∈ ((II ×_t II) Cn 𝐶))
35	1, 2, 3, 4, 5, 6, 7	cvmlift2lem7 35623	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐾) = 𝐺)
36		0elunit 13470	. . . . 5 ⊢ 0 ∈ (0[,]1)
37	1, 2, 3, 4, 5, 6, 7	cvmlift2lem8 35624	. . . . 5 ⊢ ((𝜑 ∧ 0 ∈ (0[,]1)) → (0𝐾0) = (𝐻‘0))
38	36, 37	mpan2 701	. . . 4 ⊢ (𝜑 → (0𝐾0) = (𝐻‘0))
39	1, 2, 3, 4, 5, 6	cvmlift2lem2 35618	. . . . 5 ⊢ (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐻) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝐻‘0) = 𝑃))
40	39	simp3d 1156	. . . 4 ⊢ (𝜑 → (𝐻‘0) = 𝑃)
41	38, 40	eqtrd 2796	. . 3 ⊢ (𝜑 → (0𝐾0) = 𝑃)
42		coeq2 5828	. . . . . 6 ⊢ (𝑔 = 𝐾 → (𝐹 ∘ 𝑔) = (𝐹 ∘ 𝐾))
43	42	eqeq1d 2763	. . . . 5 ⊢ (𝑔 = 𝐾 → ((𝐹 ∘ 𝑔) = 𝐺 ↔ (𝐹 ∘ 𝐾) = 𝐺))
44		oveq 7398	. . . . . 6 ⊢ (𝑔 = 𝐾 → (0𝑔0) = (0𝐾0))
45	44	eqeq1d 2763	. . . . 5 ⊢ (𝑔 = 𝐾 → ((0𝑔0) = 𝑃 ↔ (0𝐾0) = 𝑃))
46	43, 45	anbi12d 641	. . . 4 ⊢ (𝑔 = 𝐾 → (((𝐹 ∘ 𝑔) = 𝐺 ∧ (0𝑔0) = 𝑃) ↔ ((𝐹 ∘ 𝐾) = 𝐺 ∧ (0𝐾0) = 𝑃)))
47	46	rspcev 3581	. . 3 ⊢ ((𝐾 ∈ ((II ×_t II) Cn 𝐶) ∧ ((𝐹 ∘ 𝐾) = 𝐺 ∧ (0𝐾0) = 𝑃)) → ∃𝑔 ∈ ((II ×_t II) Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (0𝑔0) = 𝑃))
48	34, 35, 41, 47	syl12anc 847	. 2 ⊢ (𝜑 → ∃𝑔 ∈ ((II ×_t II) Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (0𝑔0) = 𝑃))
49		iitop 24922	. . . . 5 ⊢ II ∈ Top
50		iiuni 24923	. . . . 5 ⊢ (0[,]1) = ∪ II
51	49, 49, 50, 50	txunii 23633	. . . 4 ⊢ ((0[,]1) × (0[,]1)) = ∪ (II ×_t II)
52		iiconn 24929	. . . . . 6 ⊢ II ∈ Conn
53		txconn 23729	. . . . . 6 ⊢ ((II ∈ Conn ∧ II ∈ Conn) → (II ×_t II) ∈ Conn)
54	52, 52, 53	mp2an 702	. . . . 5 ⊢ (II ×_t II) ∈ Conn
55	54	a1i 11	. . . 4 ⊢ (𝜑 → (II ×_t II) ∈ Conn)
56		iinllyconn 35568	. . . . . 6 ⊢ II ∈ 𝑛-Locally Conn
57		txconn 23729	. . . . . . 7 ⊢ ((𝑥 ∈ Conn ∧ 𝑦 ∈ Conn) → (𝑥 ×_t 𝑦) ∈ Conn)
58	57	txnlly 23677	. . . . . 6 ⊢ ((II ∈ 𝑛-Locally Conn ∧ II ∈ 𝑛-Locally Conn) → (II ×_t II) ∈ 𝑛-Locally Conn)
59	56, 56, 58	mp2an 702	. . . . 5 ⊢ (II ×_t II) ∈ 𝑛-Locally Conn
60	59	a1i 11	. . . 4 ⊢ (𝜑 → (II ×_t II) ∈ 𝑛-Locally Conn)
61		opelxpi 5682	. . . . . 6 ⊢ ((0 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → ⟨0, 0⟩ ∈ ((0[,]1) × (0[,]1)))
62	36, 36, 61	mp2an 702	. . . . 5 ⊢ ⟨0, 0⟩ ∈ ((0[,]1) × (0[,]1))
63	62	a1i 11	. . . 4 ⊢ (𝜑 → ⟨0, 0⟩ ∈ ((0[,]1) × (0[,]1)))
64		df-ov 7395	. . . . 5 ⊢ (0𝐺0) = (𝐺‘⟨0, 0⟩)
65	5, 64	eqtrdi 2812	. . . 4 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘⟨0, 0⟩))
66	1, 51, 2, 55, 60, 63, 3, 4, 65	cvmliftmo 35598	. . 3 ⊢ (𝜑 → ∃*𝑔 ∈ ((II ×_t II) Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘⟨0, 0⟩) = 𝑃))
67		df-ov 7395	. . . . . 6 ⊢ (0𝑔0) = (𝑔‘⟨0, 0⟩)
68	67	eqeq1i 2766	. . . . 5 ⊢ ((0𝑔0) = 𝑃 ↔ (𝑔‘⟨0, 0⟩) = 𝑃)
69	68	anbi2i 632	. . . 4 ⊢ (((𝐹 ∘ 𝑔) = 𝐺 ∧ (0𝑔0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘⟨0, 0⟩) = 𝑃))
70	69	rmobii 3374	. . 3 ⊢ (∃𝑔 ∈ ((II ×_t II) Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (0𝑔0) = 𝑃) ↔ ∃𝑔 ∈ ((II ×_t II) Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘⟨0, 0⟩) = 𝑃))
71	66, 70	sylibr 236	. 2 ⊢ (𝜑 → ∃*𝑔 ∈ ((II ×_t II) Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (0𝑔0) = 𝑃))
72		reu5 3368	. 2 ⊢ (∃!𝑔 ∈ ((II ×_t II) Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (0𝑔0) = 𝑃) ↔ (∃𝑔 ∈ ((II ×_t II) Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (0𝑔0) = 𝑃) ∧ ∃*𝑔 ∈ ((II ×_t II) Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (0𝑔0) = 𝑃)))
73	48, 71, 72	sylanbrc 592	1 ⊢ (𝜑 → ∃!𝑔 ∈ ((II ×_t II) Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (0𝑔0) = 𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ∃wrex 3085 ∃!wreu 3364 ∃*wrmo 3365 {crab 3413 ⊆ wss 3904 {csn 4581 ⟨cop 4587 ∪ cuni 4864 {copab 5161 ↦ cmpt 5180 × cxp 5643 ∘ ccom 5649 ‘cfv 6517 ℩crio 7348 (class class class)co 7392 ∈ cmpo 7394 0cc0 11070 1c1 11071 [,]cicc 13349 neicnei 23137 Cn ccn 23264 CnP ccnp 23265 Conncconn 23451 𝑛-Locally cnlly 23505 ×_t ctx 23600 IIcii 24917 CovMap ccvm 35569

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 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-inf2 9593 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148 ax-addf 11149

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 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8673 df-ec 8675 df-map 8805 df-ixp 8876 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9455 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-z 12566 df-dec 12686 df-uz 12837 df-q 12947 df-rp 12991 df-xneg 13111 df-xadd 13112 df-xmul 13113 df-ioo 13350 df-ico 13352 df-icc 13353 df-fz 13510 df-fzo 13657 df-fl 13799 df-seq 14012 df-exp 14072 df-hash 14341 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246 df-clim 15498 df-sum 15697 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17515 df-qtop 17520 df-imas 17521 df-xps 17523 df-mre 17597 df-mrc 17598 df-acs 17600 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-submnd 18801 df-mulg 19093 df-cntz 19340 df-cmn 19805 df-psmet 21396 df-xmet 21397 df-met 21398 df-bl 21399 df-mopn 21400 df-cnfld 21405 df-top 22934 df-topon 22951 df-topsp 22973 df-bases 22986 df-cld 23059 df-ntr 23060 df-cls 23061 df-nei 23138 df-cn 23267 df-cnp 23268 df-cmp 23427 df-conn 23452 df-lly 23506 df-nlly 23507 df-tx 23602 df-hmeo 23795 df-xms 24360 df-ms 24361 df-tms 24362 df-ii 24919 df-cncf 24920 df-htpy 25012 df-phtpy 25013 df-phtpc 25034 df-pconn 35535 df-sconn 35536 df-cvm 35570

This theorem is referenced by: cvmlift2 35630

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