Theorem cvmlift3lem2 35375

Description: Lemma for cvmlift2 35371. (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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → 𝐾 ∈ SConn)
3		sconnpconn 35282	. . . 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 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → 𝑂 ∈ 𝑌)
7		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → 𝑋 ∈ 𝑌)
8		cvmlift3.y	. . . 4 ⊢ 𝑌 = ∪ 𝐾
9	8	pconncn 35279	. . 3 ⊢ ((𝐾 ∈ PConn ∧ 𝑂 ∈ 𝑌 ∧ 𝑋 ∈ 𝑌) → ∃𝑎 ∈ (II Cn 𝐾)((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))
10	4, 6, 7, 9	syl3anc 1373	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → ∃𝑎 ∈ (II Cn 𝐾)((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))
11		cvmlift3.b	. . . . . . . . 9 ⊢ 𝐵 = ∪ 𝐶
12		eqid 2733	. . . . . . . . 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 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
15		simprl 770	. . . . . . . . . 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 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝐺 ∈ (𝐾 Cn 𝐽))
18		cnco 23191	. . . . . . . . . 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 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝑃 ∈ 𝐵)
22		simprrl 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝑎‘0) = 𝑂)
23	22	fveq2d 6835	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐺‘(𝑎‘0)) = (𝐺‘𝑂))
24		iiuni 24811	. . . . . . . . . . . . 13 ⊢ (0[,]1) = ∪ II
25	24, 8	cnf 23171	. . . . . . . . . . . 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 13379	. . . . . . . . . . 11 ⊢ 0 ∈ (0[,]1)
28		fvco3 6930	. . . . . . . . . . 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 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐹‘𝑃) = (𝐺‘𝑂))
32	23, 29, 31	3eqtr4rd 2779	. . . . . . . . 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 35356	. . . . . . . 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 1142	. . . . . . 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 23171	. . . . . . 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 13380	. . . . . 6 ⊢ 1 ∈ (0[,]1)
38		ffvelcdm 7023	. . . . . 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 781	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝑎‘1) = 𝑋)
41		eqidd 2734	. . . . . 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 6830	. . . . . . . . 9 ⊢ (𝑓 = 𝑎 → (𝑓‘0) = (𝑎‘0))
43	42	eqeq1d 2735	. . . . . . . 8 ⊢ (𝑓 = 𝑎 → ((𝑓‘0) = 𝑂 ↔ (𝑎‘0) = 𝑂))
44		fveq1 6830	. . . . . . . . 9 ⊢ (𝑓 = 𝑎 → (𝑓‘1) = (𝑎‘1))
45	44	eqeq1d 2735	. . . . . . . 8 ⊢ (𝑓 = 𝑎 → ((𝑓‘1) = 𝑋 ↔ (𝑎‘1) = 𝑋))
46		coeq2 5805	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑎 → (𝐺 ∘ 𝑓) = (𝐺 ∘ 𝑎))
47	46	eqeq2d 2744	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑎 → ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎)))
48	47	anbi1d 631	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑎 → (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃)))
49	48	riotabidv 7314	. . . . . . . . . 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 6833	. . . . . . . . 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 2735	. . . . . . . 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 1438	. . . . . . 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 3574	. . . . . 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 1374	. . . . 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 732	. . . . . . . . 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 732	. . . . . . . . 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 732	. . . . . . . . 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 732	. . . . . . . . 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 732	. . . . . . . . 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 732	. . . . . . . . 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 732	. . . . . . . . 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 726	. . . . . . . . 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 726	. . . . . . . . 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 770	. . . . . . . . 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 1222	. . . . . . . . 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 726	. . . . . . . . . 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 1223	. . . . . . . . . 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 2771	. . . . . . . . 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 35374	. . . . . . . 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 1224	. . . . . . . 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 2768	. . . . . . 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 3136	. . . . . 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 3126	. . . . 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 2745	. . . . . . . . 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 1444	. . . . . . . 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 3158	. . . . . . 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 2737	. . . . . . . . 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 340	. . . . . . . 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 3157	. . . . . . 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 632	. . . . . 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 3574	. . . . 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 836	. . . 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 6830	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (𝑓‘0) = (ℎ‘0))
85	84	eqeq1d 2735	. . . . . . . 8 ⊢ (𝑓 = ℎ → ((𝑓‘0) = 𝑂 ↔ (ℎ‘0) = 𝑂))
86		fveq1 6830	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (𝑓‘1) = (ℎ‘1))
87	86	eqeq1d 2735	. . . . . . . 8 ⊢ (𝑓 = ℎ → ((𝑓‘1) = 𝑋 ↔ (ℎ‘1) = 𝑋))
88		coeq2 5805	. . . . . . . . . . . . 13 ⊢ (𝑓 = ℎ → (𝐺 ∘ 𝑓) = (𝐺 ∘ ℎ))
89	88	eqeq2d 2744	. . . . . . . . . . . 12 ⊢ (𝑓 = ℎ → ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ)))
90	89	anbi1d 631	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃)))
91	90	riotabidv 7314	. . . . . . . . . 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 6833	. . . . . . . . 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 2735	. . . . . . . 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 1438	. . . . . . 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	cbvrexvw 3213	. . . . . 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 2745	. . . . . . . 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 1444	. . . . . . 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 3158	. . . . . 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	bitrid 283	. . . . 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 3689	. . . 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 234	. . 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 3136	. 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 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  ∃!wreu 3346  ∪ cuni 4860   ∘ ccom 5625  ⟶wf 6485  ‘cfv 6489  ℩crio 7311  (class class class)co 7355  0cc0 11016  1c1 11017  [,]cicc 13258   Cn ccn 23149  𝑛-Locally cnlly 23390  IIcii 24805  PConncpconn 35274  SConncsconn 35275   CovMap ccvm 35310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-inf2 9541  ax-cnex 11072  ax-resscn 11073  ax-1cn 11074  ax-icn 11075  ax-addcl 11076  ax-addrcl 11077  ax-mulcl 11078  ax-mulrcl 11079  ax-mulcom 11080  ax-addass 11081  ax-mulass 11082  ax-distr 11083  ax-i2m1 11084  ax-1ne0 11085  ax-1rid 11086  ax-rnegex 11087  ax-rrecex 11088  ax-cnre 11089  ax-pre-lttri 11090  ax-pre-lttrn 11091  ax-pre-ltadd 11092  ax-pre-mulgt0 11093  ax-pre-sup 11094  ax-addf 11095

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-iin 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-of 7619  df-om 7806  df-1st 7930  df-2nd 7931  df-supp 8100  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-er 8631  df-ec 8633  df-map 8761  df-ixp 8831  df-en 8879  df-dom 8880  df-sdom 8881  df-fin 8882  df-fsupp 9256  df-fi 9305  df-sup 9336  df-inf 9337  df-oi 9406  df-card 9842  df-pnf 11158  df-mnf 11159  df-xr 11160  df-ltxr 11161  df-le 11162  df-sub 11356  df-neg 11357  df-div 11785  df-nn 12136  df-2 12198  df-3 12199  df-4 12200  df-5 12201  df-6 12202  df-7 12203  df-8 12204  df-9 12205  df-n0 12392  df-z 12479  df-dec 12599  df-uz 12743  df-q 12857  df-rp 12901  df-xneg 13021  df-xadd 13022  df-xmul 13023  df-ioo 13259  df-ico 13261  df-icc 13262  df-fz 13418  df-fzo 13565  df-fl 13706  df-seq 13919  df-exp 13979  df-hash 14248  df-cj 15016  df-re 15017  df-im 15018  df-sqrt 15152  df-abs 15153  df-clim 15405  df-sum 15604  df-struct 17068  df-sets 17085  df-slot 17103  df-ndx 17115  df-base 17131  df-ress 17152  df-plusg 17184  df-mulr 17185  df-starv 17186  df-sca 17187  df-vsca 17188  df-ip 17189  df-tset 17190  df-ple 17191  df-ds 17193  df-unif 17194  df-hom 17195  df-cco 17196  df-rest 17336  df-topn 17337  df-0g 17355  df-gsum 17356  df-topgen 17357  df-pt 17358  df-prds 17361  df-xrs 17416  df-qtop 17421  df-imas 17422  df-xps 17424  df-mre 17498  df-mrc 17499  df-acs 17501  df-mgm 18558  df-sgrp 18637  df-mnd 18653  df-submnd 18702  df-mulg 18991  df-cntz 19239  df-cmn 19704  df-psmet 21293  df-xmet 21294  df-met 21295  df-bl 21296  df-mopn 21297  df-cnfld 21302  df-top 22819  df-topon 22836  df-topsp 22858  df-bases 22871  df-cld 22944  df-ntr 22945  df-cls 22946  df-nei 23023  df-cn 23152  df-cnp 23153  df-cmp 23312  df-conn 23337  df-lly 23391  df-nlly 23392  df-tx 23487  df-hmeo 23680  df-xms 24245  df-ms 24246  df-tms 24247  df-ii 24807  df-cncf 24808  df-htpy 24906  df-phtpy 24907  df-phtpc 24928  df-pco 24942  df-pconn 35276  df-sconn 35277  df-cvm 35311

This theorem is referenced by:  cvmlift3lem3  35376  cvmlift3lem4  35377