Theorem cvmlift3lem2 35523

Description: Lemma for cvmlift2 35519. (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 35430	. . . 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 35427	. . 3 ⊢ ((𝐾 ∈ PConn ∧ 𝑂 ∈ 𝑌 ∧ 𝑋 ∈ 𝑌) → ∃𝑎 ∈ (II Cn 𝐾)((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))
10	4, 6, 7, 9	syl3anc 1374	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → ∃𝑎 ∈ (II Cn 𝐾)((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))
11		cvmlift3.b	. . . . . . . . 9 ⊢ 𝐵 = ∪ 𝐶
12		eqid 2737	. . . . . . . . 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 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
15		simprl 771	. . . . . . . . . 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 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝐺 ∈ (𝐾 Cn 𝐽))
18		cnco 23240	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺 ∘ 𝑎) ∈ (II Cn 𝐽))
19	15, 17, 18	syl2anc 585	. . . . . . . . 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 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝑃 ∈ 𝐵)
22		simprrl 781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝑎‘0) = 𝑂)
23	22	fveq2d 6836	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐺‘(𝑎‘0)) = (𝐺‘𝑂))
24		iiuni 24857	. . . . . . . . . . . . 13 ⊢ (0[,]1) = ∪ II
25	24, 8	cnf 23220	. . . . . . . . . . . 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 13411	. . . . . . . . . . 11 ⊢ 0 ∈ (0[,]1)
28		fvco3 6931	. . . . . . . . . . 11 ⊢ ((𝑎:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ 𝑎)‘0) = (𝐺‘(𝑎‘0)))
29	26, 27, 28	sylancl 587	. . . . . . . . . 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 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐹‘𝑃) = (𝐺‘𝑂))
32	23, 29, 31	3eqtr4rd 2783	. . . . . . . . 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 35504	. . . . . . . 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 1143	. . . . . . 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 23220	. . . . . . 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 13412	. . . . . 6 ⊢ 1 ∈ (0[,]1)
38		ffvelcdm 7025	. . . . . 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 587	. . . . 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 782	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝑎‘1) = 𝑋)
41		eqidd 2738	. . . . . 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 6831	. . . . . . . . 9 ⊢ (𝑓 = 𝑎 → (𝑓‘0) = (𝑎‘0))
43	42	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑓 = 𝑎 → ((𝑓‘0) = 𝑂 ↔ (𝑎‘0) = 𝑂))
44		fveq1 6831	. . . . . . . . 9 ⊢ (𝑓 = 𝑎 → (𝑓‘1) = (𝑎‘1))
45	44	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑓 = 𝑎 → ((𝑓‘1) = 𝑋 ↔ (𝑎‘1) = 𝑋))
46		coeq2 5805	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑎 → (𝐺 ∘ 𝑓) = (𝐺 ∘ 𝑎))
47	46	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑎 → ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎)))
48	47	anbi1d 632	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑎 → (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑎) ∧ (𝑔‘0) = 𝑃)))
49	48	riotabidv 7317	. . . . . . . . . 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 6834	. . . . . . . . 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 2739	. . . . . . . 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 1439	. . . . . . 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 3565	. . . . . 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 1375	. . . . 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 733	. . . . . . . . 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 733	. . . . . . . . 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 733	. . . . . . . . 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 733	. . . . . . . . 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 733	. . . . . . . . 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 733	. . . . . . . . 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 733	. . . . . . . . 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 727	. . . . . . . . 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 727	. . . . . . . . 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 771	. . . . . . . . 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 1223	. . . . . . . . 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 727	. . . . . . . . . 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 1224	. . . . . . . . . 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 2775	. . . . . . . . 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 35522	. . . . . . . 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 1225	. . . . . . . 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 2772	. . . . . . 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 3140	. . . . . 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 3130	. . . . 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 2749	. . . . . . . . 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 1445	. . . . . . . 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 3162	. . . . . . 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 2741	. . . . . . . . 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 3161	. . . . . . 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 633	. . . . . 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 3565	. . . . 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 837	. . . 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 6831	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (𝑓‘0) = (ℎ‘0))
85	84	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑓 = ℎ → ((𝑓‘0) = 𝑂 ↔ (ℎ‘0) = 𝑂))
86		fveq1 6831	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (𝑓‘1) = (ℎ‘1))
87	86	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑓 = ℎ → ((𝑓‘1) = 𝑋 ↔ (ℎ‘1) = 𝑋))
88		coeq2 5805	. . . . . . . . . . . . 13 ⊢ (𝑓 = ℎ → (𝐺 ∘ 𝑓) = (𝐺 ∘ ℎ))
89	88	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑓 = ℎ → ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ)))
90	89	anbi1d 632	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ ℎ) ∧ (𝑔‘0) = 𝑃)))
91	90	riotabidv 7317	. . . . . . . . . 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 6834	. . . . . . . . 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 2739	. . . . . . . 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 1439	. . . . . . 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 3217	. . . . . 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 2749	. . . . . . . 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 1445	. . . . . . 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 3162	. . . . . 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 3680	. . . 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 3140	. 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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  ∃!wreu 3341  ∪ cuni 4851   ∘ ccom 5626  ⟶wf 6486  ‘cfv 6490  ℩crio 7314  (class class class)co 7358  0cc0 11027  1c1 11028  [,]cicc 13290   Cn ccn 23198  𝑛-Locally cnlly 23439  IIcii 24851  PConncpconn 35422  SConncsconn 35423   CovMap ccvm 35458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-inf2 9551  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104  ax-pre-sup 11105  ax-addf 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8102  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-er 8634  df-ec 8636  df-map 8766  df-ixp 8837  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-fsupp 9266  df-fi 9315  df-sup 9346  df-inf 9347  df-oi 9416  df-card 9852  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-div 11797  df-nn 12164  df-2 12233  df-3 12234  df-4 12235  df-5 12236  df-6 12237  df-7 12238  df-8 12239  df-9 12240  df-n0 12427  df-z 12514  df-dec 12634  df-uz 12778  df-q 12888  df-rp 12932  df-xneg 13052  df-xadd 13053  df-xmul 13054  df-ioo 13291  df-ico 13293  df-icc 13294  df-fz 13451  df-fzo 13598  df-fl 13740  df-seq 13953  df-exp 14013  df-hash 14282  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15439  df-sum 15638  df-struct 17106  df-sets 17123  df-slot 17141  df-ndx 17153  df-base 17169  df-ress 17190  df-plusg 17222  df-mulr 17223  df-starv 17224  df-sca 17225  df-vsca 17226  df-ip 17227  df-tset 17228  df-ple 17229  df-ds 17231  df-unif 17232  df-hom 17233  df-cco 17234  df-rest 17374  df-topn 17375  df-0g 17393  df-gsum 17394  df-topgen 17395  df-pt 17396  df-prds 17399  df-xrs 17455  df-qtop 17460  df-imas 17461  df-xps 17463  df-mre 17537  df-mrc 17538  df-acs 17540  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-submnd 18741  df-mulg 19033  df-cntz 19281  df-cmn 19746  df-psmet 21334  df-xmet 21335  df-met 21336  df-bl 21337  df-mopn 21338  df-cnfld 21343  df-top 22868  df-topon 22885  df-topsp 22907  df-bases 22920  df-cld 22993  df-ntr 22994  df-cls 22995  df-nei 23072  df-cn 23201  df-cnp 23202  df-cmp 23361  df-conn 23386  df-lly 23440  df-nlly 23441  df-tx 23536  df-hmeo 23729  df-xms 24294  df-ms 24295  df-tms 24296  df-ii 24853  df-cncf 24854  df-htpy 24946  df-phtpy 24947  df-phtpc 24968  df-pco 24981  df-pconn 35424  df-sconn 35425  df-cvm 35459

This theorem is referenced by:  cvmlift3lem3  35524  cvmlift3lem4  35525