Theorem cvmlift3lem2 35495

Description: Lemma for cvmlift2 35491. (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 35402	. . . 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 35399	. . 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 23214	. . . . . . . . . 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 6839	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐺‘(𝑎‘0)) = (𝐺‘𝑂))
24		iiuni 24834	. . . . . . . . . . . . 13 ⊢ (0[,]1) = ∪ II
25	24, 8	cnf 23194	. . . . . . . . . . . 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 13389	. . . . . . . . . . 11 ⊢ 0 ∈ (0[,]1)
28		fvco3 6934	. . . . . . . . . . 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 35476	. . . . . . . 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 23194	. . . . . . 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 13390	. . . . . 6 ⊢ 1 ∈ (0[,]1)
38		ffvelcdm 7028	. . . . . 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 6834	. . . . . . . . 9 ⊢ (𝑓 = 𝑎 → (𝑓‘0) = (𝑎‘0))
43	42	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑓 = 𝑎 → ((𝑓‘0) = 𝑂 ↔ (𝑎‘0) = 𝑂))
44		fveq1 6834	. . . . . . . . 9 ⊢ (𝑓 = 𝑎 → (𝑓‘1) = (𝑎‘1))
45	44	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑓 = 𝑎 → ((𝑓‘1) = 𝑋 ↔ (𝑎‘1) = 𝑋))
46		coeq2 5808	. . . . . . . . . . . . 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 7319	. . . . . . . . . 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 6837	. . . . . . . . 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 3577	. . . . . 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 35494	. . . . . . . 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 3139	. . . . . 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 3129	. . . . 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 3161	. . . . . . 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 3160	. . . . . . 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 3577	. . . . 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 6834	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (𝑓‘0) = (ℎ‘0))
85	84	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑓 = ℎ → ((𝑓‘0) = 𝑂 ↔ (ℎ‘0) = 𝑂))
86		fveq1 6834	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (𝑓‘1) = (ℎ‘1))
87	86	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑓 = ℎ → ((𝑓‘1) = 𝑋 ↔ (ℎ‘1) = 𝑋))
88		coeq2 5808	. . . . . . . . . . . . 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 7319	. . . . . . . . . 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 6837	. . . . . . . . 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 3216	. . . . . 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 3161	. . . . . 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 3692	. . . 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 3139	. 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 3061  ∃!wreu 3349  ∪ cuni 4864   ∘ ccom 5629  ⟶wf 6489  ‘cfv 6493  ℩crio 7316  (class class class)co 7360  0cc0 11030  1c1 11031  [,]cicc 13268   Cn ccn 23172  𝑛-Locally cnlly 23413  IIcii 24828  PConncpconn 35394  SConncsconn 35395   CovMap ccvm 35430

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 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-inf2 9554  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108  ax-addf 11109

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 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-ec 8639  df-map 8769  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-fi 9318  df-sup 9349  df-inf 9350  df-oi 9419  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-9 12219  df-n0 12406  df-z 12493  df-dec 12612  df-uz 12756  df-q 12866  df-rp 12910  df-xneg 13030  df-xadd 13031  df-xmul 13032  df-ioo 13269  df-ico 13271  df-icc 13272  df-fz 13428  df-fzo 13575  df-fl 13716  df-seq 13929  df-exp 13989  df-hash 14258  df-cj 15026  df-re 15027  df-im 15028  df-sqrt 15162  df-abs 15163  df-clim 15415  df-sum 15614  df-struct 17078  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17141  df-ress 17162  df-plusg 17194  df-mulr 17195  df-starv 17196  df-sca 17197  df-vsca 17198  df-ip 17199  df-tset 17200  df-ple 17201  df-ds 17203  df-unif 17204  df-hom 17205  df-cco 17206  df-rest 17346  df-topn 17347  df-0g 17365  df-gsum 17366  df-topgen 17367  df-pt 17368  df-prds 17371  df-xrs 17427  df-qtop 17432  df-imas 17433  df-xps 17435  df-mre 17509  df-mrc 17510  df-acs 17512  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-submnd 18713  df-mulg 19002  df-cntz 19250  df-cmn 19715  df-psmet 21305  df-xmet 21306  df-met 21307  df-bl 21308  df-mopn 21309  df-cnfld 21314  df-top 22842  df-topon 22859  df-topsp 22881  df-bases 22894  df-cld 22967  df-ntr 22968  df-cls 22969  df-nei 23046  df-cn 23175  df-cnp 23176  df-cmp 23335  df-conn 23360  df-lly 23414  df-nlly 23415  df-tx 23510  df-hmeo 23703  df-xms 24268  df-ms 24269  df-tms 24270  df-ii 24830  df-cncf 24831  df-htpy 24929  df-phtpy 24930  df-phtpc 24951  df-pco 24965  df-pconn 35396  df-sconn 35397  df-cvm 35431

This theorem is referenced by:  cvmlift3lem3  35496  cvmlift3lem4  35497