Theorem cvmlift3lem9 32687

Description: Lemma for cvmlift2 32676. (Contributed by Mario Carneiro, 7-May-2015.)

Hypotheses

Ref	Expression
cvmlift3.b	⊢ 𝐵 = ∪ 𝐶
cvmlift3.y	⊢ 𝑌 = ∪ 𝐾
cvmlift3.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmlift3.k	⊢ (𝜑 → 𝐾 ∈ SConn)
cvmlift3.l	⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)
cvmlift3.o	⊢ (𝜑 → 𝑂 ∈ 𝑌)
cvmlift3.g	⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
cvmlift3.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
cvmlift3.e	⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))
cvmlift3.h	⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
cvmlift3lem7.s	⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})

Assertion

Ref	Expression
cvmlift3lem9	⊢ (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))

Distinct variable groups:   𝑐,𝑑,𝑓,𝑘,𝑠,𝑧,𝑔,𝑥   𝐽,𝑐   𝑔,𝑑,𝑥,𝐽,𝑓,𝑘,𝑠   𝐹,𝑐,𝑑,𝑓,𝑔,𝑘,𝑠   𝑥,𝑧,𝐹   𝐻,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧   𝑆,𝑓,𝑥   𝐵,𝑑,𝑓,𝑔,𝑥,𝑧   𝐺,𝑐,𝑑,𝑓,𝑔,𝑘,𝑥,𝑧   𝐶,𝑐,𝑑,𝑓,𝑔,𝑘,𝑠,𝑥,𝑧   𝜑,𝑓,𝑥   𝐾,𝑐,𝑓,𝑔,𝑥,𝑧   𝑃,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧   𝑂,𝑐,𝑓,𝑔,𝑥,𝑧   𝑓,𝑌,𝑔,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑧,𝑔,𝑘,𝑠,𝑐,𝑑)   𝐵(𝑘,𝑠,𝑐)   𝑃(𝑘,𝑠)   𝑆(𝑧,𝑔,𝑘,𝑠,𝑐,𝑑)   𝐺(𝑠)   𝐻(𝑘,𝑠)   𝐽(𝑧)   𝐾(𝑘,𝑠,𝑑)   𝑂(𝑘,𝑠,𝑑)   𝑌(𝑘,𝑠,𝑐,𝑑)

Proof of Theorem cvmlift3lem9

Step	Hyp	Ref	Expression
1		cvmlift3.b	. . 3 ⊢ 𝐵 = ∪ 𝐶
2		cvmlift3.y	. . 3 ⊢ 𝑌 = ∪ 𝐾
3		cvmlift3.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
4		cvmlift3.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ SConn)
5		cvmlift3.l	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)
6		cvmlift3.o	. . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑌)
7		cvmlift3.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
8		cvmlift3.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
9		cvmlift3.e	. . 3 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))
10		cvmlift3.h	. . 3 ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
11		cvmlift3lem7.s	. . 3 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	cvmlift3lem8 32686	. 2 ⊢ (𝜑 → 𝐻 ∈ (𝐾 Cn 𝐶))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	cvmlift3lem5 32683	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐻) = 𝐺)
14		iitopon 23484	. . . . . 6 ⊢ II ∈ (TopOn‘(0[,]1))
15	14	a1i 11	. . . . 5 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
16		sconntop 32588	. . . . . . 7 ⊢ (𝐾 ∈ SConn → 𝐾 ∈ Top)
17	4, 16	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Top)
18	2	toptopon 21522	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
19	17, 18	sylib 221	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
20		cnconst2 21888	. . . . 5 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑂 ∈ 𝑌) → ((0[,]1) × {𝑂}) ∈ (II Cn 𝐾))
21	15, 19, 6, 20	syl3anc 1368	. . . 4 ⊢ (𝜑 → ((0[,]1) × {𝑂}) ∈ (II Cn 𝐾))
22		0elunit 12847	. . . . 5 ⊢ 0 ∈ (0[,]1)
23		fvconst2g 6941	. . . . 5 ⊢ ((𝑂 ∈ 𝑌 ∧ 0 ∈ (0[,]1)) → (((0[,]1) × {𝑂})‘0) = 𝑂)
24	6, 22, 23	sylancl 589	. . . 4 ⊢ (𝜑 → (((0[,]1) × {𝑂})‘0) = 𝑂)
25		1elunit 12848	. . . . 5 ⊢ 1 ∈ (0[,]1)
26		fvconst2g 6941	. . . . 5 ⊢ ((𝑂 ∈ 𝑌 ∧ 1 ∈ (0[,]1)) → (((0[,]1) × {𝑂})‘1) = 𝑂)
27	6, 25, 26	sylancl 589	. . . 4 ⊢ (𝜑 → (((0[,]1) × {𝑂})‘1) = 𝑂)
28	9	sneqd 4537	. . . . . . . . 9 ⊢ (𝜑 → {(𝐹‘𝑃)} = {(𝐺‘𝑂)})
29	28	xpeq2d 5549	. . . . . . . 8 ⊢ (𝜑 → ((0[,]1) × {(𝐹‘𝑃)}) = ((0[,]1) × {(𝐺‘𝑂)}))
30		cvmcn 32622	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
31		eqid 2798	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
32	1, 31	cnf 21851	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶∪ 𝐽)
33		ffn 6487	. . . . . . . . . 10 ⊢ (𝐹:𝐵⟶∪ 𝐽 → 𝐹 Fn 𝐵)
34	3, 30, 32, 33	4syl 19	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝐵)
35		fcoconst 6873	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐵 ∧ 𝑃 ∈ 𝐵) → (𝐹 ∘ ((0[,]1) × {𝑃})) = ((0[,]1) × {(𝐹‘𝑃)}))
36	34, 8, 35	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ ((0[,]1) × {𝑃})) = ((0[,]1) × {(𝐹‘𝑃)}))
37	2, 31	cnf 21851	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌⟶∪ 𝐽)
38	7, 37	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:𝑌⟶∪ 𝐽)
39	38	ffnd 6488	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 Fn 𝑌)
40		fcoconst 6873	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝑌 ∧ 𝑂 ∈ 𝑌) → (𝐺 ∘ ((0[,]1) × {𝑂})) = ((0[,]1) × {(𝐺‘𝑂)}))
41	39, 6, 40	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ∘ ((0[,]1) × {𝑂})) = ((0[,]1) × {(𝐺‘𝑂)}))
42	29, 36, 41	3eqtr4d 2843	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ ((0[,]1) × {𝑃})) = (𝐺 ∘ ((0[,]1) × {𝑂})))
43		fvconst2g 6941	. . . . . . . 8 ⊢ ((𝑃 ∈ 𝐵 ∧ 0 ∈ (0[,]1)) → (((0[,]1) × {𝑃})‘0) = 𝑃)
44	8, 22, 43	sylancl 589	. . . . . . 7 ⊢ (𝜑 → (((0[,]1) × {𝑃})‘0) = 𝑃)
45		cvmtop1 32620	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
46	3, 45	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ Top)
47	1	toptopon 21522	. . . . . . . . . 10 ⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
48	46, 47	sylib 221	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (TopOn‘𝐵))
49		cnconst2 21888	. . . . . . . . 9 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐶 ∈ (TopOn‘𝐵) ∧ 𝑃 ∈ 𝐵) → ((0[,]1) × {𝑃}) ∈ (II Cn 𝐶))
50	15, 48, 8, 49	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → ((0[,]1) × {𝑃}) ∈ (II Cn 𝐶))
51		cvmtop2 32621	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
52	3, 51	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ Top)
53	31	toptopon 21522	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
54	52, 53	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽))
55	38, 6	ffvelrnd 6829	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺‘𝑂) ∈ ∪ 𝐽)
56		cnconst2 21888	. . . . . . . . . . 11 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (𝐺‘𝑂) ∈ ∪ 𝐽) → ((0[,]1) × {(𝐺‘𝑂)}) ∈ (II Cn 𝐽))
57	15, 54, 55, 56	syl3anc 1368	. . . . . . . . . 10 ⊢ (𝜑 → ((0[,]1) × {(𝐺‘𝑂)}) ∈ (II Cn 𝐽))
58	41, 57	eqeltrd 2890	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ∘ ((0[,]1) × {𝑂})) ∈ (II Cn 𝐽))
59		fvconst2g 6941	. . . . . . . . . . 11 ⊢ (((𝐺‘𝑂) ∈ ∪ 𝐽 ∧ 0 ∈ (0[,]1)) → (((0[,]1) × {(𝐺‘𝑂)})‘0) = (𝐺‘𝑂))
60	55, 22, 59	sylancl 589	. . . . . . . . . 10 ⊢ (𝜑 → (((0[,]1) × {(𝐺‘𝑂)})‘0) = (𝐺‘𝑂))
61	41	fveq1d 6647	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐺 ∘ ((0[,]1) × {𝑂}))‘0) = (((0[,]1) × {(𝐺‘𝑂)})‘0))
62	60, 61, 9	3eqtr4rd 2844	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑃) = ((𝐺 ∘ ((0[,]1) × {𝑂}))‘0))
63	1	cvmlift 32659	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝐺 ∘ ((0[,]1) × {𝑂})) ∈ (II Cn 𝐽)) ∧ (𝑃 ∈ 𝐵 ∧ (𝐹‘𝑃) = ((𝐺 ∘ ((0[,]1) × {𝑂}))‘0))) → ∃!𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃))
64	3, 58, 8, 62, 63	syl22anc 837	. . . . . . . 8 ⊢ (𝜑 → ∃!𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃))
65		coeq2 5693	. . . . . . . . . . 11 ⊢ (𝑔 = ((0[,]1) × {𝑃}) → (𝐹 ∘ 𝑔) = (𝐹 ∘ ((0[,]1) × {𝑃})))
66	65	eqeq1d 2800	. . . . . . . . . 10 ⊢ (𝑔 = ((0[,]1) × {𝑃}) → ((𝐹 ∘ 𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ↔ (𝐹 ∘ ((0[,]1) × {𝑃})) = (𝐺 ∘ ((0[,]1) × {𝑂}))))
67		fveq1 6644	. . . . . . . . . . 11 ⊢ (𝑔 = ((0[,]1) × {𝑃}) → (𝑔‘0) = (((0[,]1) × {𝑃})‘0))
68	67	eqeq1d 2800	. . . . . . . . . 10 ⊢ (𝑔 = ((0[,]1) × {𝑃}) → ((𝑔‘0) = 𝑃 ↔ (((0[,]1) × {𝑃})‘0) = 𝑃))
69	66, 68	anbi12d 633	. . . . . . . . 9 ⊢ (𝑔 = ((0[,]1) × {𝑃}) → (((𝐹 ∘ 𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ ((0[,]1) × {𝑃})) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (((0[,]1) × {𝑃})‘0) = 𝑃)))
70	69	riota2 7118	. . . . . . . 8 ⊢ ((((0[,]1) × {𝑃}) ∈ (II Cn 𝐶) ∧ ∃!𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃)) → (((𝐹 ∘ ((0[,]1) × {𝑃})) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (((0[,]1) × {𝑃})‘0) = 𝑃) ↔ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃)) = ((0[,]1) × {𝑃})))
71	50, 64, 70	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → (((𝐹 ∘ ((0[,]1) × {𝑃})) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (((0[,]1) × {𝑃})‘0) = 𝑃) ↔ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃)) = ((0[,]1) × {𝑃})))
72	42, 44, 71	mpbi2and 711	. . . . . 6 ⊢ (𝜑 → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃)) = ((0[,]1) × {𝑃}))
73	72	fveq1d 6647	. . . . 5 ⊢ (𝜑 → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃))‘1) = (((0[,]1) × {𝑃})‘1))
74		fvconst2g 6941	. . . . . 6 ⊢ ((𝑃 ∈ 𝐵 ∧ 1 ∈ (0[,]1)) → (((0[,]1) × {𝑃})‘1) = 𝑃)
75	8, 25, 74	sylancl 589	. . . . 5 ⊢ (𝜑 → (((0[,]1) × {𝑃})‘1) = 𝑃)
76	73, 75	eqtrd 2833	. . . 4 ⊢ (𝜑 → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃)
77		fveq1 6644	. . . . . . 7 ⊢ (𝑓 = ((0[,]1) × {𝑂}) → (𝑓‘0) = (((0[,]1) × {𝑂})‘0))
78	77	eqeq1d 2800	. . . . . 6 ⊢ (𝑓 = ((0[,]1) × {𝑂}) → ((𝑓‘0) = 𝑂 ↔ (((0[,]1) × {𝑂})‘0) = 𝑂))
79		fveq1 6644	. . . . . . 7 ⊢ (𝑓 = ((0[,]1) × {𝑂}) → (𝑓‘1) = (((0[,]1) × {𝑂})‘1))
80	79	eqeq1d 2800	. . . . . 6 ⊢ (𝑓 = ((0[,]1) × {𝑂}) → ((𝑓‘1) = 𝑂 ↔ (((0[,]1) × {𝑂})‘1) = 𝑂))
81		coeq2 5693	. . . . . . . . . . 11 ⊢ (𝑓 = ((0[,]1) × {𝑂}) → (𝐺 ∘ 𝑓) = (𝐺 ∘ ((0[,]1) × {𝑂})))
82	81	eqeq2d 2809	. . . . . . . . . 10 ⊢ (𝑓 = ((0[,]1) × {𝑂}) → ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂}))))
83	82	anbi1d 632	. . . . . . . . 9 ⊢ (𝑓 = ((0[,]1) × {𝑂}) → (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃)))
84	83	riotabidv 7095	. . . . . . . 8 ⊢ (𝑓 = ((0[,]1) × {𝑂}) → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃)))
85	84	fveq1d 6647	. . . . . . 7 ⊢ (𝑓 = ((0[,]1) × {𝑂}) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃))‘1))
86	85	eqeq1d 2800	. . . . . 6 ⊢ (𝑓 = ((0[,]1) × {𝑂}) → (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃 ↔ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃))
87	78, 80, 86	3anbi123d 1433	. . . . 5 ⊢ (𝑓 = ((0[,]1) × {𝑂}) → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑂 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃) ↔ ((((0[,]1) × {𝑂})‘0) = 𝑂 ∧ (((0[,]1) × {𝑂})‘1) = 𝑂 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃)))
88	87	rspcev 3571	. . . 4 ⊢ ((((0[,]1) × {𝑂}) ∈ (II Cn 𝐾) ∧ ((((0[,]1) × {𝑂})‘0) = 𝑂 ∧ (((0[,]1) × {𝑂})‘1) = 𝑂 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ((0[,]1) × {𝑂})) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃)) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑂 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃))
89	21, 24, 27, 76, 88	syl13anc 1369	. . 3 ⊢ (𝜑 → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑂 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃))
90	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	cvmlift3lem4 32682	. . . 4 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝑌) → ((𝐻‘𝑂) = 𝑃 ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑂 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃)))
91	6, 90	mpdan 686	. . 3 ⊢ (𝜑 → ((𝐻‘𝑂) = 𝑃 ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑂 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑃)))
92	89, 91	mpbird 260	. 2 ⊢ (𝜑 → (𝐻‘𝑂) = 𝑃)
93		coeq2 5693	. . . . 5 ⊢ (𝑓 = 𝐻 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝐻))
94	93	eqeq1d 2800	. . . 4 ⊢ (𝑓 = 𝐻 → ((𝐹 ∘ 𝑓) = 𝐺 ↔ (𝐹 ∘ 𝐻) = 𝐺))
95		fveq1 6644	. . . . 5 ⊢ (𝑓 = 𝐻 → (𝑓‘𝑂) = (𝐻‘𝑂))
96	95	eqeq1d 2800	. . . 4 ⊢ (𝑓 = 𝐻 → ((𝑓‘𝑂) = 𝑃 ↔ (𝐻‘𝑂) = 𝑃))
97	94, 96	anbi12d 633	. . 3 ⊢ (𝑓 = 𝐻 → (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ↔ ((𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘𝑂) = 𝑃)))
98	97	rspcev 3571	. 2 ⊢ ((𝐻 ∈ (𝐾 Cn 𝐶) ∧ ((𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘𝑂) = 𝑃)) → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))
99	12, 13, 92, 98	syl12anc 835	1 ⊢ (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  ∃!wreu 3108  {crab 3110   ∖ cdif 3878   ∩ cin 3880  ∅c0 4243  𝒫 cpw 4497  {csn 4525  ∪ cuni 4800   ↦ cmpt 5110   × cxp 5517  ^◡ccnv 5518   ↾ cres 5521   “ cima 5522   ∘ ccom 5523   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  ℩crio 7092  (class class class)co 7135  0cc0 10526  1c1 10527  [,]cicc 12729   ↾_t crest 16686  Topctop 21498  TopOnctopon 21515   Cn ccn 21829  𝑛-Locally cnlly 22070  Homeochmeo 22358  IIcii 23480  PConncpconn 32579  SConncsconn 32580   CovMap ccvm 32615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605  ax-mulf 10606

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-ec 8274  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-fi 8859  df-sup 8890  df-inf 8891  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ico 12732  df-icc 12733  df-fz 12886  df-fzo 13029  df-fl 13157  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-sum 15035  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-starv 16572  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-hom 16581  df-cco 16582  df-rest 16688  df-topn 16689  df-0g 16707  df-gsum 16708  df-topgen 16709  df-pt 16710  df-prds 16713  df-xrs 16767  df-qtop 16772  df-imas 16773  df-xps 16775  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-mulg 18217  df-cntz 18439  df-cmn 18900  df-psmet 20083  df-xmet 20084  df-met 20085  df-bl 20086  df-mopn 20087  df-cnfld 20092  df-top 21499  df-topon 21516  df-topsp 21538  df-bases 21551  df-cld 21624  df-ntr 21625  df-cls 21626  df-nei 21703  df-cn 21832  df-cnp 21833  df-cmp 21992  df-conn 22017  df-lly 22071  df-nlly 22072  df-tx 22167  df-hmeo 22360  df-xms 22927  df-ms 22928  df-tms 22929  df-ii 23482  df-htpy 23575  df-phtpy 23576  df-phtpc 23597  df-pco 23610  df-pconn 32581  df-sconn 32582  df-cvm 32616

This theorem is referenced by:  cvmlift3  32688