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Theorem cvmlift3lem2 34800
Description: Lemma for cvmlift2 34796. (Contributed by Mario Carneiro, 6-Jul-2015.)
Hypotheses
Ref Expression
cvmlift3.b 𝐵 = 𝐶
cvmlift3.y 𝑌 = 𝐾
cvmlift3.f (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
cvmlift3.k (𝜑𝐾 ∈ SConn)
cvmlift3.l (𝜑𝐾 ∈ 𝑛-Locally PConn)
cvmlift3.o (𝜑𝑂𝑌)
cvmlift3.g (𝜑𝐺 ∈ (𝐾 Cn 𝐽))
cvmlift3.p (𝜑𝑃𝐵)
cvmlift3.e (𝜑 → (𝐹𝑃) = (𝐺𝑂))
Assertion
Ref Expression
cvmlift3lem2 ((𝜑𝑋𝑌) → ∃!𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvmlift3.k . . . . 5 (𝜑𝐾 ∈ SConn)
21adantr 480 . . . 4 ((𝜑𝑋𝑌) → 𝐾 ∈ SConn)
3 sconnpconn 34707 . . . 4 (𝐾 ∈ SConn → 𝐾 ∈ PConn)
42, 3syl 17 . . 3 ((𝜑𝑋𝑌) → 𝐾 ∈ PConn)
5 cvmlift3.o . . . 4 (𝜑𝑂𝑌)
65adantr 480 . . 3 ((𝜑𝑋𝑌) → 𝑂𝑌)
7 simpr 484 . . 3 ((𝜑𝑋𝑌) → 𝑋𝑌)
8 cvmlift3.y . . . 4 𝑌 = 𝐾
98pconncn 34704 . . 3 ((𝐾 ∈ PConn ∧ 𝑂𝑌𝑋𝑌) → ∃𝑎 ∈ (II Cn 𝐾)((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))
104, 6, 7, 9syl3anc 1368 . 2 ((𝜑𝑋𝑌) → ∃𝑎 ∈ (II Cn 𝐾)((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))
11 cvmlift3.b . . . . . . . . 9 𝐵 = 𝐶
12 eqid 2724 . . . . . . . . 9 (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))
13 cvmlift3.f . . . . . . . . . 10 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
1413ad2antrr 723 . . . . . . . . 9 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
15 simprl 768 . . . . . . . . . 10 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝑎 ∈ (II Cn 𝐾))
16 cvmlift3.g . . . . . . . . . . 11 (𝜑𝐺 ∈ (𝐾 Cn 𝐽))
1716ad2antrr 723 . . . . . . . . . 10 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝐺 ∈ (𝐾 Cn 𝐽))
18 cnco 23092 . . . . . . . . . 10 ((𝑎 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺𝑎) ∈ (II Cn 𝐽))
1915, 17, 18syl2anc 583 . . . . . . . . 9 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐺𝑎) ∈ (II Cn 𝐽))
20 cvmlift3.p . . . . . . . . . 10 (𝜑𝑃𝐵)
2120ad2antrr 723 . . . . . . . . 9 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝑃𝐵)
22 simprrl 778 . . . . . . . . . . 11 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝑎‘0) = 𝑂)
2322fveq2d 6885 . . . . . . . . . 10 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐺‘(𝑎‘0)) = (𝐺𝑂))
24 iiuni 24723 . . . . . . . . . . . . 13 (0[,]1) = II
2524, 8cnf 23072 . . . . . . . . . . . 12 (𝑎 ∈ (II Cn 𝐾) → 𝑎:(0[,]1)⟶𝑌)
2615, 25syl 17 . . . . . . . . . . 11 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝑎:(0[,]1)⟶𝑌)
27 0elunit 13443 . . . . . . . . . . 11 0 ∈ (0[,]1)
28 fvco3 6980 . . . . . . . . . . 11 ((𝑎:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺𝑎)‘0) = (𝐺‘(𝑎‘0)))
2926, 27, 28sylancl 585 . . . . . . . . . 10 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → ((𝐺𝑎)‘0) = (𝐺‘(𝑎‘0)))
30 cvmlift3.e . . . . . . . . . . 11 (𝜑 → (𝐹𝑃) = (𝐺𝑂))
3130ad2antrr 723 . . . . . . . . . 10 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐹𝑃) = (𝐺𝑂))
3223, 29, 313eqtr4rd 2775 . . . . . . . . 9 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐹𝑃) = ((𝐺𝑎)‘0))
3311, 12, 14, 19, 21, 32cvmliftiota 34781 . . . . . . . 8 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶) ∧ (𝐹 ∘ (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))) = (𝐺𝑎) ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘0) = 𝑃))
3433simp1d 1139 . . . . . . 7 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶))
3524, 11cnf 23072 . . . . . . 7 ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶) → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵)
3634, 35syl 17 . . . . . 6 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵)
37 1elunit 13444 . . . . . 6 1 ∈ (0[,]1)
38 ffvelcdm 7073 . . . . . 6 (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) ∈ 𝐵)
3936, 37, 38sylancl 585 . . . . 5 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) ∈ 𝐵)
40 simprrr 779 . . . . . 6 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝑎‘1) = 𝑋)
41 eqidd 2725 . . . . . 6 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1))
42 fveq1 6880 . . . . . . . . 9 (𝑓 = 𝑎 → (𝑓‘0) = (𝑎‘0))
4342eqeq1d 2726 . . . . . . . 8 (𝑓 = 𝑎 → ((𝑓‘0) = 𝑂 ↔ (𝑎‘0) = 𝑂))
44 fveq1 6880 . . . . . . . . 9 (𝑓 = 𝑎 → (𝑓‘1) = (𝑎‘1))
4544eqeq1d 2726 . . . . . . . 8 (𝑓 = 𝑎 → ((𝑓‘1) = 𝑋 ↔ (𝑎‘1) = 𝑋))
46 coeq2 5848 . . . . . . . . . . . . 13 (𝑓 = 𝑎 → (𝐺𝑓) = (𝐺𝑎))
4746eqeq2d 2735 . . . . . . . . . . . 12 (𝑓 = 𝑎 → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺𝑎)))
4847anbi1d 629 . . . . . . . . . . 11 (𝑓 = 𝑎 → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃)))
4948riotabidv 7359 . . . . . . . . . 10 (𝑓 = 𝑎 → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃)))
5049fveq1d 6883 . . . . . . . . 9 (𝑓 = 𝑎 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1))
5150eqeq1d 2726 . . . . . . . 8 (𝑓 = 𝑎 → (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) ↔ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1)))
5243, 45, 513anbi123d 1432 . . . . . . 7 (𝑓 = 𝑎 → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1)) ↔ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1))))
5352rspcev 3604 . . . . . 6 ((𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1))) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1)))
5415, 22, 40, 41, 53syl13anc 1369 . . . . 5 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1)))
5513ad4antr 729 . . . . . . . . 9 (((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) ∧ ( ∈ (II Cn 𝐾) ∧ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
561ad4antr 729 . . . . . . . . 9 (((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) ∧ ( ∈ (II Cn 𝐾) ∧ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → 𝐾 ∈ SConn)
57 cvmlift3.l . . . . . . . . . 10 (𝜑𝐾 ∈ 𝑛-Locally PConn)
5857ad4antr 729 . . . . . . . . 9 (((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) ∧ ( ∈ (II Cn 𝐾) ∧ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → 𝐾 ∈ 𝑛-Locally PConn)
595ad4antr 729 . . . . . . . . 9 (((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) ∧ ( ∈ (II Cn 𝐾) ∧ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → 𝑂𝑌)
6016ad4antr 729 . . . . . . . . 9 (((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) ∧ ( ∈ (II Cn 𝐾) ∧ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → 𝐺 ∈ (𝐾 Cn 𝐽))
6120ad4antr 729 . . . . . . . . 9 (((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) ∧ ( ∈ (II Cn 𝐾) ∧ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → 𝑃𝐵)
6230ad4antr 729 . . . . . . . . 9 (((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) ∧ ( ∈ (II Cn 𝐾) ∧ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → (𝐹𝑃) = (𝐺𝑂))
6315ad2antrr 723 . . . . . . . . 9 (((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) ∧ ( ∈ (II Cn 𝐾) ∧ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → 𝑎 ∈ (II Cn 𝐾))
6422ad2antrr 723 . . . . . . . . 9 (((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) ∧ ( ∈ (II Cn 𝐾) ∧ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → (𝑎‘0) = 𝑂)
65 simprl 768 . . . . . . . . 9 (((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) ∧ ( ∈ (II Cn 𝐾) ∧ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → ∈ (II Cn 𝐾))
66 simprr1 1218 . . . . . . . . 9 (((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) ∧ ( ∈ (II Cn 𝐾) ∧ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → (‘0) = 𝑂)
6740ad2antrr 723 . . . . . . . . . 10 (((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) ∧ ( ∈ (II Cn 𝐾) ∧ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → (𝑎‘1) = 𝑋)
68 simprr2 1219 . . . . . . . . . 10 (((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) ∧ ( ∈ (II Cn 𝐾) ∧ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → (‘1) = 𝑋)
6967, 68eqtr4d 2767 . . . . . . . . 9 (((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) ∧ ( ∈ (II Cn 𝐾) ∧ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → (𝑎‘1) = (‘1))
7011, 8, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 69cvmlift3lem1 34799 . . . . . . . 8 (((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) ∧ ( ∈ (II Cn 𝐾) ∧ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1))
71 simprr3 1220 . . . . . . . 8 (((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) ∧ ( ∈ (II Cn 𝐾) ∧ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤)
7270, 71eqtrd 2764 . . . . . . 7 (((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) ∧ ( ∈ (II Cn 𝐾) ∧ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤)
7372rexlimdvaa 3148 . . . . . 6 ((((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) ∧ 𝑤𝐵) → (∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))
7473ralrimiva 3138 . . . . 5 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → ∀𝑤𝐵 (∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))
75 eqeq2 2736 . . . . . . . . 9 (𝑧 = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) → (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧 ↔ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1)))
76753anbi3d 1438 . . . . . . . 8 (𝑧 = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1))))
7776rexbidv 3170 . . . . . . 7 (𝑧 = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1))))
78 eqeq1 2728 . . . . . . . . 9 (𝑧 = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) → (𝑧 = 𝑤 ↔ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))
7978imbi2d 340 . . . . . . . 8 (𝑧 = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) → ((∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → 𝑧 = 𝑤) ↔ (∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤)))
8079ralbidv 3169 . . . . . . 7 (𝑧 = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) → (∀𝑤𝐵 (∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → 𝑧 = 𝑤) ↔ ∀𝑤𝐵 (∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤)))
8177, 80anbi12d 630 . . . . . 6 (𝑧 = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) → ((∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ∧ ∀𝑤𝐵 (∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → 𝑧 = 𝑤)) ↔ (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1)) ∧ ∀𝑤𝐵 (∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))))
8281rspcev 3604 . . . . 5 ((((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) ∈ 𝐵 ∧ (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1)) ∧ ∀𝑤𝐵 (∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))) → ∃𝑧𝐵 (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ∧ ∀𝑤𝐵 (∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → 𝑧 = 𝑤)))
8339, 54, 74, 82syl12anc 834 . . . 4 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → ∃𝑧𝐵 (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ∧ ∀𝑤𝐵 (∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → 𝑧 = 𝑤)))
84 fveq1 6880 . . . . . . . . 9 (𝑓 = → (𝑓‘0) = (‘0))
8584eqeq1d 2726 . . . . . . . 8 (𝑓 = → ((𝑓‘0) = 𝑂 ↔ (‘0) = 𝑂))
86 fveq1 6880 . . . . . . . . 9 (𝑓 = → (𝑓‘1) = (‘1))
8786eqeq1d 2726 . . . . . . . 8 (𝑓 = → ((𝑓‘1) = 𝑋 ↔ (‘1) = 𝑋))
88 coeq2 5848 . . . . . . . . . . . . 13 (𝑓 = → (𝐺𝑓) = (𝐺))
8988eqeq2d 2735 . . . . . . . . . . . 12 (𝑓 = → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺)))
9089anbi1d 629 . . . . . . . . . . 11 (𝑓 = → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
9190riotabidv 7359 . . . . . . . . . 10 (𝑓 = → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
9291fveq1d 6883 . . . . . . . . 9 (𝑓 = → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1))
9392eqeq1d 2726 . . . . . . . 8 (𝑓 = → (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧 ↔ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))
9485, 87, 933anbi123d 1432 . . . . . . 7 (𝑓 = → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
9594cbvrexvw 3227 . . . . . 6 (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))
96 eqeq2 2736 . . . . . . . 8 (𝑧 = 𝑤 → (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧 ↔ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤))
97963anbi3d 1438 . . . . . . 7 (𝑧 = 𝑤 → (((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤)))
9897rexbidv 3170 . . . . . 6 (𝑧 = 𝑤 → (∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤)))
9995, 98bitrid 283 . . . . 5 (𝑧 = 𝑤 → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤)))
10099reu8 3721 . . . 4 (∃!𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ∃𝑧𝐵 (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ∧ ∀𝑤𝐵 (∃ ∈ (II Cn 𝐾)((‘0) = 𝑂 ∧ (‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑤) → 𝑧 = 𝑤)))
10183, 100sylibr 233 . . 3 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → ∃!𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))
102101rexlimdvaa 3148 . 2 ((𝜑𝑋𝑌) → (∃𝑎 ∈ (II Cn 𝐾)((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋) → ∃!𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
10310, 102mpd 15 1 ((𝜑𝑋𝑌) → ∃!𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3053  wrex 3062  ∃!wreu 3366   cuni 4899  ccom 5670  wf 6529  cfv 6533  crio 7356  (class class class)co 7401  0cc0 11106  1c1 11107  [,]cicc 13324   Cn ccn 23050  𝑛-Locally cnlly 23291  IIcii 24717  PConncpconn 34699  SConncsconn 34700   CovMap ccvm 34735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184  ax-addf 11185
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-of 7663  df-om 7849  df-1st 7968  df-2nd 7969  df-supp 8141  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-er 8699  df-ec 8701  df-map 8818  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-fi 9402  df-sup 9433  df-inf 9434  df-oi 9501  df-card 9930  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-q 12930  df-rp 12972  df-xneg 13089  df-xadd 13090  df-xmul 13091  df-ioo 13325  df-ico 13327  df-icc 13328  df-fz 13482  df-fzo 13625  df-fl 13754  df-seq 13964  df-exp 14025  df-hash 14288  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-clim 15429  df-sum 15630  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-starv 17211  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-hom 17220  df-cco 17221  df-rest 17367  df-topn 17368  df-0g 17386  df-gsum 17387  df-topgen 17388  df-pt 17389  df-prds 17392  df-xrs 17447  df-qtop 17452  df-imas 17453  df-xps 17455  df-mre 17529  df-mrc 17530  df-acs 17532  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-submnd 18704  df-mulg 18986  df-cntz 19223  df-cmn 19692  df-psmet 21220  df-xmet 21221  df-met 21222  df-bl 21223  df-mopn 21224  df-cnfld 21229  df-top 22718  df-topon 22735  df-topsp 22757  df-bases 22771  df-cld 22845  df-ntr 22846  df-cls 22847  df-nei 22924  df-cn 23053  df-cnp 23054  df-cmp 23213  df-conn 23238  df-lly 23292  df-nlly 23293  df-tx 23388  df-hmeo 23581  df-xms 24148  df-ms 24149  df-tms 24150  df-ii 24719  df-cncf 24720  df-htpy 24818  df-phtpy 24819  df-phtpc 24840  df-pco 24854  df-pconn 34701  df-sconn 34702  df-cvm 34736
This theorem is referenced by:  cvmlift3lem3  34801  cvmlift3lem4  34802
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