Theorem cvmlift3lem4 35307

Description: Lemma for cvmlift2 35301. (Contributed by Mario Carneiro, 6-Jul-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)))

Assertion

Ref	Expression
cvmlift3lem4	⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → ((𝐻‘𝑋) = 𝐴 ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴)))

Distinct variable groups:   𝑧,𝑓,𝐴   𝑓,𝑔,𝑧,𝑥   𝑓,𝐽   𝑥,𝑔,𝐽   𝑓,𝐹,𝑔   𝑥,𝑧,𝐹   𝑓,𝐻,𝑔,𝑥,𝑧   𝐵,𝑓,𝑔,𝑥,𝑧   𝑓,𝑋,𝑔,𝑥,𝑧   𝑓,𝐺,𝑔,𝑥,𝑧   𝐶,𝑓,𝑔,𝑥,𝑧   𝜑,𝑓,𝑥   𝑓,𝐾,𝑔,𝑥,𝑧   𝑃,𝑓,𝑔,𝑥,𝑧   𝑓,𝑂,𝑔,𝑥,𝑧   𝑓,𝑌,𝑔,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑧,𝑔)   𝐴(𝑥,𝑔)   𝐽(𝑧)

Proof of Theorem cvmlift3lem4

Step	Hyp	Ref	Expression
1		cvmlift3.b	. . . . 5 ⊢ 𝐵 = ∪ 𝐶
2		cvmlift3.y	. . . . 5 ⊢ 𝑌 = ∪ 𝐾
3		cvmlift3.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
4		cvmlift3.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ SConn)
5		cvmlift3.l	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)
6		cvmlift3.o	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ 𝑌)
7		cvmlift3.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
8		cvmlift3.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
9		cvmlift3.e	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))
10		cvmlift3.h	. . . . 5 ⊢ 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	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	cvmlift3lem3 35306	. . . 4 ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)
12	11	ffvelcdmda 7104	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑋) ∈ 𝐵)
13		eleq1 2827	. . 3 ⊢ ((𝐻‘𝑋) = 𝐴 → ((𝐻‘𝑋) ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
14	12, 13	syl5ibcom 245	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → ((𝐻‘𝑋) = 𝐴 → 𝐴 ∈ 𝐵))
15		eqid 2735	. . . . . . . . . . 11 ⊢ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))
16	3	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
17		simprl 771	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → 𝑓 ∈ (II Cn 𝐾))
18	7	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → 𝐺 ∈ (𝐾 Cn 𝐽))
19		cnco 23290	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺 ∘ 𝑓) ∈ (II Cn 𝐽))
20	17, 18, 19	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝐺 ∘ 𝑓) ∈ (II Cn 𝐽))
21	8	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → 𝑃 ∈ 𝐵)
22		simprr 773	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝑓‘0) = 𝑂)
23	22	fveq2d 6911	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝐺‘(𝑓‘0)) = (𝐺‘𝑂))
24		iiuni 24921	. . . . . . . . . . . . . . 15 ⊢ (0[,]1) = ∪ II
25	24, 2	cnf 23270	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (II Cn 𝐾) → 𝑓:(0[,]1)⟶𝑌)
26	17, 25	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → 𝑓:(0[,]1)⟶𝑌)
27		0elunit 13506	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0[,]1)
28		fvco3 7008	. . . . . . . . . . . . 13 ⊢ ((𝑓:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ 𝑓)‘0) = (𝐺‘(𝑓‘0)))
29	26, 27, 28	sylancl 586	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → ((𝐺 ∘ 𝑓)‘0) = (𝐺‘(𝑓‘0)))
30	9	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝐹‘𝑃) = (𝐺‘𝑂))
31	23, 29, 30	3eqtr4rd 2786	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝐹‘𝑃) = ((𝐺 ∘ 𝑓)‘0))
32	1, 15, 16, 20, 21, 31	cvmliftiota 35286	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶) ∧ (𝐹 ∘ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))) = (𝐺 ∘ 𝑓) ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘0) = 𝑃))
33	32	simp1d 1141	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶))
34	24, 1	cnf 23270	. . . . . . . . 9 ⊢ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶) → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵)
35	33, 34	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵)
36		1elunit 13507	. . . . . . . 8 ⊢ 1 ∈ (0[,]1)
37		ffvelcdm 7101	. . . . . . . 8 ⊢ (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) ∈ 𝐵)
38	35, 36, 37	sylancl 586	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) ∈ 𝐵)
39		eleq1 2827	. . . . . . 7 ⊢ (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴 → (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
40	38, 39	syl5ibcom 245	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴 → 𝐴 ∈ 𝐵))
41	40	expr 456	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) → ((𝑓‘0) = 𝑂 → (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴 → 𝐴 ∈ 𝐵)))
42	41	a1dd 50	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) → ((𝑓‘0) = 𝑂 → ((𝑓‘1) = 𝑋 → (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴 → 𝐴 ∈ 𝐵))))
43	42	3impd 1347	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴) → 𝐴 ∈ 𝐵))
44	43	rexlimdva 3153	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴) → 𝐴 ∈ 𝐵))
45		eqeq2 2747	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → ((𝑓‘1) = 𝑥 ↔ (𝑓‘1) = 𝑋))
46	45	3anbi2d 1440	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
47	46	rexbidv 3177	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
48	47	riotabidv 7390	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (℩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)) = (℩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)))
49		riotaex 7392	. . . . . . . 8 ⊢ (℩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)) ∈ V
50	48, 10, 49	fvmpt 7016	. . . . . . 7 ⊢ (𝑋 ∈ 𝑌 → (𝐻‘𝑋) = (℩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)))
51	50	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑋) = (℩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)))
52	51	eqeq1d 2737	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → ((𝐻‘𝑋) = 𝐴 ↔ (℩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)) = 𝐴))
53	52	adantl 481	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝜑 ∧ 𝑋 ∈ 𝑌)) → ((𝐻‘𝑋) = 𝐴 ↔ (℩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)) = 𝐴))
54	1, 2, 3, 4, 5, 6, 7, 8, 9	cvmlift3lem2 35305	. . . . 5 ⊢ ((𝜑 ∧ 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))
55		eqeq2 2747	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧 ↔ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴))
56	55	3anbi3d 1441	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴)))
57	56	rexbidv 3177	. . . . . 6 ⊢ (𝑧 = 𝐴 → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴)))
58	57	riota2 7413	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃!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)) → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴) ↔ (℩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)) = 𝐴))
59	54, 58	sylan2 593	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝜑 ∧ 𝑋 ∈ 𝑌)) → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴) ↔ (℩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)) = 𝐴))
60	53, 59	bitr4d 282	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝜑 ∧ 𝑋 ∈ 𝑌)) → ((𝐻‘𝑋) = 𝐴 ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴)))
61	60	expcom 413	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → (𝐴 ∈ 𝐵 → ((𝐻‘𝑋) = 𝐴 ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴))))
62	14, 44, 61	pm5.21ndd 379	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → ((𝐻‘𝑋) = 𝐴 ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∃wrex 3068  ∃!wreu 3376  ∪ cuni 4912   ↦ cmpt 5231   ∘ ccom 5693  ⟶wf 6559  ‘cfv 6563  ℩crio 7387  (class class class)co 7431  0cc0 11153  1c1 11154  [,]cicc 13387   Cn ccn 23248  𝑛-Locally cnlly 23489  IIcii 24915  PConncpconn 35204  SConncsconn 35205   CovMap ccvm 35240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-ec 8746  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-cn 23251  df-cnp 23252  df-cmp 23411  df-conn 23436  df-lly 23490  df-nlly 23491  df-tx 23586  df-hmeo 23779  df-xms 24346  df-ms 24347  df-tms 24348  df-ii 24917  df-cncf 24918  df-htpy 25016  df-phtpy 25017  df-phtpc 25038  df-pco 25052  df-pconn 35206  df-sconn 35207  df-cvm 35241

This theorem is referenced by:  cvmlift3lem5  35308  cvmlift3lem6  35309  cvmlift3lem7  35310  cvmlift3lem9  35312