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Theorem cvmlift3lem2 35325
Description: Lemma for cvmlift2 35321. (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 35232 . . . 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 35229 . . 3 ((𝐾 ∈ PConn ∧ 𝑂𝑌𝑋𝑌) → ∃𝑎 ∈ (II Cn 𝐾)((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))
104, 6, 7, 9syl3anc 1373 . 2 ((𝜑𝑋𝑌) → ∃𝑎 ∈ (II Cn 𝐾)((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))
11 cvmlift3.b . . . . . . . . 9 𝐵 = 𝐶
12 eqid 2737 . . . . . . . . 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 726 . . . . . . . . 9 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
15 simprl 771 . . . . . . . . . 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 726 . . . . . . . . . 10 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝐺 ∈ (𝐾 Cn 𝐽))
18 cnco 23274 . . . . . . . . . 10 ((𝑎 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺𝑎) ∈ (II Cn 𝐽))
1915, 17, 18syl2anc 584 . . . . . . . . 9 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐺𝑎) ∈ (II Cn 𝐽))
20 cvmlift3.p . . . . . . . . . 10 (𝜑𝑃𝐵)
2120ad2antrr 726 . . . . . . . . 9 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝑃𝐵)
22 simprrl 781 . . . . . . . . . . 11 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝑎‘0) = 𝑂)
2322fveq2d 6910 . . . . . . . . . 10 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐺‘(𝑎‘0)) = (𝐺𝑂))
24 iiuni 24907 . . . . . . . . . . . . 13 (0[,]1) = II
2524, 8cnf 23254 . . . . . . . . . . . 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 13509 . . . . . . . . . . 11 0 ∈ (0[,]1)
28 fvco3 7008 . . . . . . . . . . 11 ((𝑎:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺𝑎)‘0) = (𝐺‘(𝑎‘0)))
2926, 27, 28sylancl 586 . . . . . . . . . 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 726 . . . . . . . . . 10 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐹𝑃) = (𝐺𝑂))
3223, 29, 313eqtr4rd 2788 . . . . . . . . 9 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐹𝑃) = ((𝐺𝑎)‘0))
3311, 12, 14, 19, 21, 32cvmliftiota 35306 . . . . . . . 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 1143 . . . . . . 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 23254 . . . . . . 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 13510 . . . . . 6 1 ∈ (0[,]1)
38 ffvelcdm 7101 . . . . . 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 586 . . . . 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 782 . . . . . 6 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝑎‘1) = 𝑋)
41 eqidd 2738 . . . . . 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 6905 . . . . . . . . 9 (𝑓 = 𝑎 → (𝑓‘0) = (𝑎‘0))
4342eqeq1d 2739 . . . . . . . 8 (𝑓 = 𝑎 → ((𝑓‘0) = 𝑂 ↔ (𝑎‘0) = 𝑂))
44 fveq1 6905 . . . . . . . . 9 (𝑓 = 𝑎 → (𝑓‘1) = (𝑎‘1))
4544eqeq1d 2739 . . . . . . . 8 (𝑓 = 𝑎 → ((𝑓‘1) = 𝑋 ↔ (𝑎‘1) = 𝑋))
46 coeq2 5869 . . . . . . . . . . . . 13 (𝑓 = 𝑎 → (𝐺𝑓) = (𝐺𝑎))
4746eqeq2d 2748 . . . . . . . . . . . 12 (𝑓 = 𝑎 → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺𝑎)))
4847anbi1d 631 . . . . . . . . . . 11 (𝑓 = 𝑎 → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃)))
4948riotabidv 7390 . . . . . . . . . 10 (𝑓 = 𝑎 → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃)))
5049fveq1d 6908 . . . . . . . . 9 (𝑓 = 𝑎 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1))
5150eqeq1d 2739 . . . . . . . 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 1438 . . . . . . 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 3622 . . . . . 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 1374 . . . . 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 732 . . . . . . . . 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 732 . . . . . . . . 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 732 . . . . . . . . 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 732 . . . . . . . . 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 732 . . . . . . . . 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 732 . . . . . . . . 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 732 . . . . . . . . 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 726 . . . . . . . . 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 726 . . . . . . . . 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 771 . . . . . . . . 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 1222 . . . . . . . . 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 726 . . . . . . . . . 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 1223 . . . . . . . . . 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 2780 . . . . . . . . 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 35324 . . . . . . . 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 1224 . . . . . . . 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 2777 . . . . . . 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 3156 . . . . . 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 3146 . . . . 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 2749 . . . . . . . . 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 1444 . . . . . . . 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 3179 . . . . . . 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 2741 . . . . . . . . 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 3178 . . . . . . 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 632 . . . . . 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 3622 . . . . 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 837 . . . 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 6905 . . . . . . . . 9 (𝑓 = → (𝑓‘0) = (‘0))
8584eqeq1d 2739 . . . . . . . 8 (𝑓 = → ((𝑓‘0) = 𝑂 ↔ (‘0) = 𝑂))
86 fveq1 6905 . . . . . . . . 9 (𝑓 = → (𝑓‘1) = (‘1))
8786eqeq1d 2739 . . . . . . . 8 (𝑓 = → ((𝑓‘1) = 𝑋 ↔ (‘1) = 𝑋))
88 coeq2 5869 . . . . . . . . . . . . 13 (𝑓 = → (𝐺𝑓) = (𝐺))
8988eqeq2d 2748 . . . . . . . . . . . 12 (𝑓 = → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺)))
9089anbi1d 631 . . . . . . . . . . 11 (𝑓 = → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
9190riotabidv 7390 . . . . . . . . . 10 (𝑓 = → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
9291fveq1d 6908 . . . . . . . . 9 (𝑓 = → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1))
9392eqeq1d 2739 . . . . . . . 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 1438 . . . . . . 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 3238 . . . . . 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 2749 . . . . . . . 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 1444 . . . . . . 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 3179 . . . . . 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 3739 . . . 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 234 . . 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 3156 . 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 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  ∃!wreu 3378   cuni 4907  ccom 5689  wf 6557  cfv 6561  crio 7387  (class class class)co 7431  0cc0 11155  1c1 11156  [,]cicc 13390   Cn ccn 23232  𝑛-Locally cnlly 23473  IIcii 24901  PConncpconn 35224  SConncsconn 35225   CovMap ccvm 35260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-ec 8747  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-cn 23235  df-cnp 23236  df-cmp 23395  df-conn 23420  df-lly 23474  df-nlly 23475  df-tx 23570  df-hmeo 23763  df-xms 24330  df-ms 24331  df-tms 24332  df-ii 24903  df-cncf 24904  df-htpy 25002  df-phtpy 25003  df-phtpc 25024  df-pco 25038  df-pconn 35226  df-sconn 35227  df-cvm 35261
This theorem is referenced by:  cvmlift3lem3  35326  cvmlift3lem4  35327
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