Theorem cvmlift3lem4 35327

Description: Lemma for cvmlift2 35321. (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 35326	. . . 4 ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)
12	11	ffvelcdmda 7104	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑋) ∈ 𝐵)
13		eleq1 2829	. . 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 2737	. . . . . . . . . . 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 23274	. . . . . . . . . . . 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 6910	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝐺‘(𝑓‘0)) = (𝐺‘𝑂))
24		iiuni 24907	. . . . . . . . . . . . . . 15 ⊢ (0[,]1) = ∪ II
25	24, 2	cnf 23254	. . . . . . . . . . . . . 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 13509	. . . . . . . . . . . . 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 2788	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝐹‘𝑃) = ((𝐺 ∘ 𝑓)‘0))
32	1, 15, 16, 20, 21, 31	cvmliftiota 35306	. . . . . . . . . 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 1143	. . . . . . . . 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 23254	. . . . . . . . 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 13510	. . . . . . . 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 2829	. . . . . . 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 1349	. . 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 3155	. 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 2749	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → ((𝑓‘1) = 𝑥 ↔ (𝑓‘1) = 𝑋))
46	45	3anbi2d 1443	. . . . . . . . . 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 3179	. . . . . . . . 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 2739	. . . . 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 35325	. . . . 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 2749	. . . . . . . 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 1444	. . . . . . 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 3179	. . . . . 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 1087   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  ∃!wreu 3378  ∪ cuni 4907   ↦ cmpt 5225   ∘ ccom 5689  ⟶wf 6557  ‘cfv 6561  ℩crio 7387  (class class class)co 7431  0cc0 11155  1c1 11156  [,]cicc 13390   Cn ccn 23232  𝑛-Locally cnlly 23473  IIcii 24901  PConncpconn 35224  SConncsconn 35225   CovMap ccvm 35260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-ec 8747  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-cn 23235  df-cnp 23236  df-cmp 23395  df-conn 23420  df-lly 23474  df-nlly 23475  df-tx 23570  df-hmeo 23763  df-xms 24330  df-ms 24331  df-tms 24332  df-ii 24903  df-cncf 24904  df-htpy 25002  df-phtpy 25003  df-phtpc 25024  df-pco 25038  df-pconn 35226  df-sconn 35227  df-cvm 35261

This theorem is referenced by:  cvmlift3lem5  35328  cvmlift3lem6  35329  cvmlift3lem7  35330  cvmlift3lem9  35332