Theorem cvmlift3lem4 35909

Description: Lemma for cvmlift2 35903. (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 35908	. . . 4 ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)
12	11	ffvelcdmda 7081	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑋) ∈ 𝐵)
13		eleq1 2850	. . 3 ⊢ ((𝐻‘𝑋) = 𝐴 → ((𝐻‘𝑋) ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
14	12, 13	syl5ibcom 248	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → ((𝐻‘𝑋) = 𝐴 → 𝐴 ∈ 𝐵))
15		eqid 2762	. . . . . . . . . . 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 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
17		simprl 783	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → 𝑓 ∈ (II Cn 𝐾))
18	7	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → 𝐺 ∈ (𝐾 Cn 𝐽))
19		cnco 23497	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺 ∘ 𝑓) ∈ (II Cn 𝐽))
20	17, 18, 19	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝐺 ∘ 𝑓) ∈ (II Cn 𝐽))
21	8	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → 𝑃 ∈ 𝐵)
22		simprr 785	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝑓‘0) = 𝑂)
23	22	fveq2d 6886	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝐺‘(𝑓‘0)) = (𝐺‘𝑂))
24		iiuni 25115	. . . . . . . . . . . . . . 15 ⊢ (0[,]1) = ∪ II
25	24, 2	cnf 23477	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (II Cn 𝐾) → 𝑓:(0[,]1)⟶𝑌)
26	17, 25	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → 𝑓:(0[,]1)⟶𝑌)
27		0elunit 13526	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0[,]1)
28		fvco3 6982	. . . . . . . . . . . . 13 ⊢ ((𝑓:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ 𝑓)‘0) = (𝐺‘(𝑓‘0)))
29	26, 27, 28	sylancl 598	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → ((𝐺 ∘ 𝑓)‘0) = (𝐺‘(𝑓‘0)))
30	9	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝐹‘𝑃) = (𝐺‘𝑂))
31	23, 29, 30	3eqtr4rd 2808	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝐹‘𝑃) = ((𝐺 ∘ 𝑓)‘0))
32	1, 15, 16, 20, 21, 31	cvmliftiota 35888	. . . . . . . . . 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 1160	. . . . . . . . 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 23477	. . . . . . . . 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 18	. . . . . . . 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 13527	. . . . . . . 8 ⊢ 1 ∈ (0[,]1)
37		ffvelcdm 7078	. . . . . . . 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 598	. . . . . . 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 2850	. . . . . . 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 248	. . . . . 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 462	. . . . 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 51	. . . 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 1367	. . 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 3165	. 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 2774	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → ((𝑓‘1) = 𝑥 ↔ (𝑓‘1) = 𝑋))
46	45	3anbi2d 1469	. . . . . . . . . 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 3188	. . . . . . . . 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 7376	. . . . . . . 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 7378	. . . . . . . 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 6990	. . . . . . 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 487	. . . . . 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 2764	. . . . 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 487	. . . 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 35907	. . . . 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 2774	. . . . . . . 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 1470	. . . . . . 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 3188	. . . . . 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 7399	. . . . 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 605	. . . 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 285	. . 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 419	. 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 382	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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  ∃!wreu 3365  ∪ cuni 4870   ↦ cmpt 5190   ∘ ccom 5663  ⟶wf 6533  ‘cfv 6537  ℩crio 7373  (class class class)co 7417  0cc0 11128  1c1 11129  [,]cicc 13405   Cn ccn 23455  𝑛-Locally cnlly 23697  IIcii 25109  PConncpconn 35806  SConncsconn 35807   CovMap ccvm 35842

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-inf2 9624  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-pre-sup 11206  ax-addf 11207

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8163  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-ec 8702  df-map 8832  df-ixp 8909  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-fsupp 9336  df-fi 9385  df-sup 9416  df-inf 9417  df-oi 9486  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-div 11900  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-7 12336  df-8 12337  df-9 12338  df-n0 12533  df-z 12620  df-dec 12741  df-uz 12892  df-q 13002  df-rp 13047  df-xneg 13167  df-xadd 13168  df-xmul 13169  df-ioo 13406  df-ico 13408  df-icc 13409  df-fz 13566  df-fzo 13714  df-fl 13857  df-seq 14070  df-exp 14130  df-hash 14399  df-cj 15190  df-re 15191  df-im 15192  df-sqrt 15326  df-abs 15327  df-clim 15579  df-sum 15778  df-struct 17245  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-plusg 17361  df-mulr 17362  df-starv 17363  df-sca 17364  df-vsca 17365  df-ip 17366  df-tset 17367  df-ple 17368  df-ds 17370  df-unif 17371  df-hom 17372  df-cco 17373  df-rest 17513  df-topn 17514  df-0g 17532  df-gsum 17533  df-topgen 17534  df-pt 17535  df-prds 17538  df-xrs 17594  df-qtop 17599  df-imas 17600  df-xps 17602  df-mre 17676  df-mrc 17677  df-acs 17679  df-mgm 18736  df-sgrp 18827  df-mnd 18843  df-submnd 18898  df-mulg 19197  df-cntz 19450  df-cmn 19915  df-psmet 21583  df-xmet 21584  df-met 21585  df-bl 21586  df-mopn 21587  df-cnfld 21592  df-top 23125  df-topon 23142  df-topsp 23164  df-bases 23177  df-cld 23250  df-ntr 23251  df-cls 23252  df-nei 23329  df-cn 23458  df-cnp 23459  df-cmp 23618  df-conn 23643  df-lly 23698  df-nlly 23699  df-tx 23794  df-hmeo 23987  df-xms 24552  df-ms 24553  df-tms 24554  df-ii 25111  df-cncf 25112  df-htpy 25204  df-phtpy 25205  df-phtpc 25226  df-pco 25239  df-pconn 35808  df-sconn 35809  df-cvm 35843

This theorem is used by:  cvmlift3lem5  35910  cvmlift3lem6  35911  cvmlift3lem7  35912  cvmlift3lem9  35914