Theorem cvmlift3lem2 32181

Description: Lemma for cvmlift2 32177. (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))

Assertion

Ref	Expression
cvmlift3lem2	⊢ ((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))

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

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

Proof of Theorem cvmlift3lem2

Dummy variables 𝑤 𝑎 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmlift3.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ SConn)
2	1	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → 𝐾 ∈ SConn)
3		sconnpconn 32088	. . . 4 ⊢ (𝐾 ∈ SConn → 𝐾 ∈ PConn)
4	2, 3	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → 𝐾 ∈ PConn)
5		cvmlift3.o	. . . 4 ⊢ (𝜑 → 𝑂 ∈ 𝑌)
6	5	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → 𝑂 ∈ 𝑌)
7		simpr 485	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → 𝑋 ∈ 𝑌)
8		cvmlift3.y	. . . 4 ⊢ 𝑌 = ∪ 𝐾
9	8	pconncn 32085	. . 3 ⊢ ((𝐾 ∈ PConn ∧ 𝑂 ∈ 𝑌 ∧ 𝑋 ∈ 𝑌) → ∃𝑎 ∈ (II Cn 𝐾)((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))
10	4, 6, 7, 9	syl3anc 1364	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → ∃𝑎 ∈ (II Cn 𝐾)((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))
11		cvmlift3.b	. . . . . . . . 9 ⊢ 𝐵 = ∪ 𝐶
12		eqid 2795	. . . . . . . . 9 ⊢ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))
13		cvmlift3.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
14	13	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
15		simprl 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝑎 ∈ (II Cn 𝐾))
16		cvmlift3.g	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
17	16	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝐺 ∈ (𝐾 Cn 𝐽))
18		cnco 21563	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺 ∘ 𝑎) ∈ (II Cn 𝐽))
19	15, 17, 18	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐺 ∘ 𝑎) ∈ (II Cn 𝐽))
20		cvmlift3.p	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
21	20	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝑃 ∈ 𝐵)
22		simprrl 777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝑎‘0) = 𝑂)
23	22	fveq2d 6547	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐺‘(𝑎‘0)) = (𝐺‘𝑂))
24		iiuni 23177	. . . . . . . . . . . . 13 ⊢ (0[,]1) = ∪ II
25	24, 8	cnf 21543	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (II Cn 𝐾) → 𝑎:(0[,]1)⟶𝑌)
26	15, 25	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝑎:(0[,]1)⟶𝑌)
27		0elunit 12710	. . . . . . . . . . 11 ⊢ 0 ∈ (0[,]1)
28		fvco3 6632	. . . . . . . . . . 11 ⊢ ((𝑎:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ 𝑎)‘0) = (𝐺‘(𝑎‘0)))
29	26, 27, 28	sylancl 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → ((𝐺 ∘ 𝑎)‘0) = (𝐺‘(𝑎‘0)))
30		cvmlift3.e	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))
31	30	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐹‘𝑃) = (𝐺‘𝑂))
32	23, 29, 31	3eqtr4rd 2842	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐹‘𝑃) = ((𝐺 ∘ 𝑎)‘0))
33	11, 12, 14, 19, 21, 32	cvmliftiota 32162	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶) ∧ (𝐹 ∘ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))) = (𝐺 ∘ 𝑎) ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘0) = 𝑃))
34	33	simp1d 1135	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶))
35	24, 11	cnf 21543	. . . . . . 7 ⊢ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶) → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵)
36	34, 35	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵)
37		1elunit 12711	. . . . . 6 ⊢ 1 ∈ (0[,]1)
38		ffvelrn 6719	. . . . . 6 ⊢ (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) ∈ 𝐵)
39	36, 37, 38	sylancl 586	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) ∈ 𝐵)
40		simprrr 778	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝑎‘1) = 𝑋)
41		eqidd 2796	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1))
42		fveq1 6542	. . . . . . . . 9 ⊢ (𝑓 = 𝑎 → (𝑓‘0) = (𝑎‘0))
43	42	eqeq1d 2797	. . . . . . . 8 ⊢ (𝑓 = 𝑎 → ((𝑓‘0) = 𝑂 ↔ (𝑎‘0) = 𝑂))
44		fveq1 6542	. . . . . . . . 9 ⊢ (𝑓 = 𝑎 → (𝑓‘1) = (𝑎‘1))
45	44	eqeq1d 2797	. . . . . . . 8 ⊢ (𝑓 = 𝑎 → ((𝑓‘1) = 𝑋 ↔ (𝑎‘1) = 𝑋))
46		coeq2 5620	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑎 → (𝐺 ∘ 𝑓) = (𝐺 ∘ 𝑎))
47	46	eqeq2d 2805	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑎 → ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎)))
48	47	anbi1d 629	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑎 → (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃)))
49	48	riotabidv 6984	. . . . . . . . . 10 ⊢ (𝑓 = 𝑎 → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃)))
50	49	fveq1d 6545	. . . . . . . . 9 ⊢ (𝑓 = 𝑎 → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1))
51	50	eqeq1d 2797	. . . . . . . 8 ⊢ (𝑓 = 𝑎 → (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) ↔ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1)))
52	43, 45, 51	3anbi123d 1428	. . . . . . 7 ⊢ (𝑓 = 𝑎 → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1)) ↔ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1))))
53	52	rspcev 3559	. . . . . 6 ⊢ ((𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1))) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1)))
54	15, 22, 40, 41, 53	syl13anc 1365	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1)))
55	13	ad4antr 728	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) ∧ (ℎ ∈ (II Cn 𝐾) ∧ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
56	1	ad4antr 728	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) ∧ (ℎ ∈ (II Cn 𝐾) ∧ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → 𝐾 ∈ SConn)
57		cvmlift3.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)
58	57	ad4antr 728	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) ∧ (ℎ ∈ (II Cn 𝐾) ∧ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → 𝐾 ∈ 𝑛-Locally PConn)
59	5	ad4antr 728	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) ∧ (ℎ ∈ (II Cn 𝐾) ∧ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → 𝑂 ∈ 𝑌)
60	16	ad4antr 728	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) ∧ (ℎ ∈ (II Cn 𝐾) ∧ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → 𝐺 ∈ (𝐾 Cn 𝐽))
61	20	ad4antr 728	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) ∧ (ℎ ∈ (II Cn 𝐾) ∧ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → 𝑃 ∈ 𝐵)
62	30	ad4antr 728	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) ∧ (ℎ ∈ (II Cn 𝐾) ∧ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → (𝐹‘𝑃) = (𝐺‘𝑂))
63	15	ad2antrr 722	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) ∧ (ℎ ∈ (II Cn 𝐾) ∧ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → 𝑎 ∈ (II Cn 𝐾))
64	22	ad2antrr 722	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) ∧ (ℎ ∈ (II Cn 𝐾) ∧ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → (𝑎‘0) = 𝑂)
65		simprl 767	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) ∧ (ℎ ∈ (II Cn 𝐾) ∧ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → ℎ ∈ (II Cn 𝐾))
66		simprr1 1214	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) ∧ (ℎ ∈ (II Cn 𝐾) ∧ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → (ℎ‘0) = 𝑂)
67	40	ad2antrr 722	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) ∧ (ℎ ∈ (II Cn 𝐾) ∧ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → (𝑎‘1) = 𝑋)
68		simprr2 1215	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) ∧ (ℎ ∈ (II Cn 𝐾) ∧ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → (ℎ‘1) = 𝑋)
69	67, 68	eqtr4d 2834	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) ∧ (ℎ ∈ (II Cn 𝐾) ∧ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → (𝑎‘1) = (ℎ‘1))
70	11, 8, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 69	cvmlift3lem1 32180	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) ∧ (ℎ ∈ (II Cn 𝐾) ∧ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1))
71		simprr3 1216	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) ∧ (ℎ ∈ (II Cn 𝐾) ∧ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤)
72	70, 71	eqtrd 2831	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) ∧ (ℎ ∈ (II Cn 𝐾) ∧ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤)
73	72	rexlimdvaa 3248	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤 ∈ 𝐵) → (∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))
74	73	ralrimiva 3149	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → ∀𝑤 ∈ 𝐵 (∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))
75		eqeq2 2806	. . . . . . . . 9 ⊢ (𝑧 = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) → (((℩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) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1)))
76	75	3anbi3d 1434	. . . . . . . 8 ⊢ (𝑧 = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) → (((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) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1))))
77	76	rexbidv 3260	. . . . . . 7 ⊢ (𝑧 = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) → (∃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) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1))))
78		eqeq1 2799	. . . . . . . . 9 ⊢ (𝑧 = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) → (𝑧 = 𝑤 ↔ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))
79	78	imbi2d 342	. . . . . . . 8 ⊢ (𝑧 = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) → ((∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → 𝑧 = 𝑤) ↔ (∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤)))
80	79	ralbidv 3164	. . . . . . 7 ⊢ (𝑧 = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) → (∀𝑤 ∈ 𝐵 (∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤 ∈ 𝐵 (∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤)))
81	77, 80	anbi12d 630	. . . . . 6 ⊢ (𝑧 = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) → ((∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ∧ ∀𝑤 ∈ 𝐵 (∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → 𝑧 = 𝑤)) ↔ (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1)) ∧ ∀𝑤 ∈ 𝐵 (∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))))
82	81	rspcev 3559	. . . . 5 ⊢ ((((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) ∈ 𝐵 ∧ (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1)) ∧ ∀𝑤 ∈ 𝐵 (∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → ∃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) ∧ ∀𝑤 ∈ 𝐵 (∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → 𝑧 = 𝑤)))
83	39, 54, 74, 82	syl12anc 833	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 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) ∧ ∀𝑤 ∈ 𝐵 (∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → 𝑧 = 𝑤)))
84		fveq1 6542	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (𝑓‘0) = (ℎ‘0))
85	84	eqeq1d 2797	. . . . . . . 8 ⊢ (𝑓 = ℎ → ((𝑓‘0) = 𝑂 ↔ (ℎ‘0) = 𝑂))
86		fveq1 6542	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (𝑓‘1) = (ℎ‘1))
87	86	eqeq1d 2797	. . . . . . . 8 ⊢ (𝑓 = ℎ → ((𝑓‘1) = 𝑋 ↔ (ℎ‘1) = 𝑋))
88		coeq2 5620	. . . . . . . . . . . . 13 ⊢ (𝑓 = ℎ → (𝐺 ∘ 𝑓) = (𝐺 ∘ ℎ))
89	88	eqeq2d 2805	. . . . . . . . . . . 12 ⊢ (𝑓 = ℎ → ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ)))
90	89	anbi1d 629	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃)))
91	90	riotabidv 6984	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃)))
92	91	fveq1d 6545	. . . . . . . . 9 ⊢ (𝑓 = ℎ → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1))
93	92	eqeq1d 2797	. . . . . . . 8 ⊢ (𝑓 = ℎ → (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧 ↔ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))
94	85, 87, 93	3anbi123d 1428	. . . . . . 7 ⊢ (𝑓 = ℎ → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
95	94	cbvrexv 3404	. . . . . 6 ⊢ (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))
96		eqeq2 2806	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧 ↔ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))
97	96	3anbi3d 1434	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤)))
98	97	rexbidv 3260	. . . . . 6 ⊢ (𝑧 = 𝑤 → (∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤)))
99	95, 98	syl5bb 284	. . . . 5 ⊢ (𝑧 = 𝑤 → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤)))
100	99	reu8 3661	. . . 4 ⊢ (∃!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) ∧ ∀𝑤 ∈ 𝐵 (∃ℎ ∈ (II Cn 𝐾)((ℎ‘0) = 𝑂 ∧ (ℎ‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → 𝑧 = 𝑤)))
101	83, 100	sylibr 235	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 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))
102	101	rexlimdvaa 3248	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → (∃𝑎 ∈ (II Cn 𝐾)((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 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)))
103	10, 102	mpd 15	1 ⊢ ((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))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  ∀wral 3105  ∃wrex 3106  ∃!wreu 3107  ∪ cuni 4749   ∘ ccom 5452  ⟶wf 6226  ‘cfv 6230  ℩crio 6981  (class class class)co 7021  0cc0 10388  1c1 10389  [,]cicc 12596   Cn ccn 21521  𝑛-Locally cnlly 21762  IIcii 23171  PConncpconn 32080  SConncsconn 32081   CovMap ccvm 32116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-inf2 8955  ax-cnex 10444  ax-resscn 10445  ax-1cn 10446  ax-icn 10447  ax-addcl 10448  ax-addrcl 10449  ax-mulcl 10450  ax-mulrcl 10451  ax-mulcom 10452  ax-addass 10453  ax-mulass 10454  ax-distr 10455  ax-i2m1 10456  ax-1ne0 10457  ax-1rid 10458  ax-rnegex 10459  ax-rrecex 10460  ax-cnre 10461  ax-pre-lttri 10462  ax-pre-lttrn 10463  ax-pre-ltadd 10464  ax-pre-mulgt0 10465  ax-pre-sup 10466  ax-addf 10467  ax-mulf 10468

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-int 4787  df-iun 4831  df-iin 4832  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-se 5408  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-isom 6239  df-riota 6982  df-ov 7024  df-oprab 7025  df-mpo 7026  df-of 7272  df-om 7442  df-1st 7550  df-2nd 7551  df-supp 7687  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-1o 7958  df-2o 7959  df-oadd 7962  df-er 8144  df-ec 8146  df-map 8263  df-ixp 8316  df-en 8363  df-dom 8364  df-sdom 8365  df-fin 8366  df-fsupp 8685  df-fi 8726  df-sup 8757  df-inf 8758  df-oi 8825  df-card 9219  df-pnf 10528  df-mnf 10529  df-xr 10530  df-ltxr 10531  df-le 10532  df-sub 10724  df-neg 10725  df-div 11151  df-nn 11492  df-2 11553  df-3 11554  df-4 11555  df-5 11556  df-6 11557  df-7 11558  df-8 11559  df-9 11560  df-n0 11751  df-z 11835  df-dec 11953  df-uz 12099  df-q 12203  df-rp 12245  df-xneg 12362  df-xadd 12363  df-xmul 12364  df-ioo 12597  df-ico 12599  df-icc 12600  df-fz 12748  df-fzo 12889  df-fl 13017  df-seq 13225  df-exp 13285  df-hash 13546  df-cj 14297  df-re 14298  df-im 14299  df-sqrt 14433  df-abs 14434  df-clim 14684  df-sum 14882  df-struct 16319  df-ndx 16320  df-slot 16321  df-base 16323  df-sets 16324  df-ress 16325  df-plusg 16412  df-mulr 16413  df-starv 16414  df-sca 16415  df-vsca 16416  df-ip 16417  df-tset 16418  df-ple 16419  df-ds 16421  df-unif 16422  df-hom 16423  df-cco 16424  df-rest 16530  df-topn 16531  df-0g 16549  df-gsum 16550  df-topgen 16551  df-pt 16552  df-prds 16555  df-xrs 16609  df-qtop 16614  df-imas 16615  df-xps 16617  df-mre 16691  df-mrc 16692  df-acs 16694  df-mgm 17686  df-sgrp 17728  df-mnd 17739  df-submnd 17780  df-mulg 17987  df-cntz 18193  df-cmn 18640  df-psmet 20224  df-xmet 20225  df-met 20226  df-bl 20227  df-mopn 20228  df-cnfld 20233  df-top 21191  df-topon 21208  df-topsp 21230  df-bases 21243  df-cld 21316  df-ntr 21317  df-cls 21318  df-nei 21395  df-cn 21524  df-cnp 21525  df-cmp 21684  df-conn 21709  df-lly 21763  df-nlly 21764  df-tx 21859  df-hmeo 22052  df-xms 22618  df-ms 22619  df-tms 22620  df-ii 23173  df-htpy 23262  df-phtpy 23263  df-phtpc 23284  df-pco 23297  df-pconn 32082  df-sconn 32083  df-cvm 32117

This theorem is referenced by:  cvmlift3lem3  32182  cvmlift3lem4  32183