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Theorem cvmlift3lem2 31428
 Description: Lemma for cvmlift2 31424. (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 31335 . . . 4 (𝐾 ∈ SConn → 𝐾 ∈ PConn)
42, 3syl 17 . . 3 ((𝜑𝑋𝑌) → 𝐾 ∈ PConn)
5 cvmlift3.o . . . 4 (𝜑𝑂𝑌)
65adantr 480 . . 3 ((𝜑𝑋𝑌) → 𝑂𝑌)
7 simpr 476 . . 3 ((𝜑𝑋𝑌) → 𝑋𝑌)
8 cvmlift3.y . . . 4 𝑌 = 𝐾
98pconncn 31332 . . 3 ((𝐾 ∈ PConn ∧ 𝑂𝑌𝑋𝑌) → ∃𝑎 ∈ (II Cn 𝐾)((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))
104, 6, 7, 9syl3anc 1366 . 2 ((𝜑𝑋𝑌) → ∃𝑎 ∈ (II Cn 𝐾)((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))
11 cvmlift3.b . . . . . . . . 9 𝐵 = 𝐶
12 eqid 2651 . . . . . . . . 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 762 . . . . . . . . 9 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
15 simprl 809 . . . . . . . . . 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 762 . . . . . . . . . 10 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝐺 ∈ (𝐾 Cn 𝐽))
18 cnco 21118 . . . . . . . . . 10 ((𝑎 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺𝑎) ∈ (II Cn 𝐽))
1915, 17, 18syl2anc 694 . . . . . . . . 9 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐺𝑎) ∈ (II Cn 𝐽))
20 cvmlift3.p . . . . . . . . . 10 (𝜑𝑃𝐵)
2120ad2antrr 762 . . . . . . . . 9 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝑃𝐵)
22 simprrl 821 . . . . . . . . . . 11 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝑎‘0) = 𝑂)
2322fveq2d 6233 . . . . . . . . . 10 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐺‘(𝑎‘0)) = (𝐺𝑂))
24 iiuni 22731 . . . . . . . . . . . . 13 (0[,]1) = II
2524, 8cnf 21098 . . . . . . . . . . . 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 12328 . . . . . . . . . . 11 0 ∈ (0[,]1)
28 fvco3 6314 . . . . . . . . . . 11 ((𝑎:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺𝑎)‘0) = (𝐺‘(𝑎‘0)))
2926, 27, 28sylancl 695 . . . . . . . . . 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 762 . . . . . . . . . 10 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐹𝑃) = (𝐺𝑂))
3223, 29, 313eqtr4rd 2696 . . . . . . . . 9 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐹𝑃) = ((𝐺𝑎)‘0))
3311, 12, 14, 19, 21, 32cvmliftiota 31409 . . . . . . . 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 1093 . . . . . . 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 21098 . . . . . . 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 12329 . . . . . 6 1 ∈ (0[,]1)
38 ffvelrn 6397 . . . . . 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 695 . . . . 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 822 . . . . . 6 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝑎‘1) = 𝑋)
41 eqidd 2652 . . . . . 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 6228 . . . . . . . . 9 (𝑓 = 𝑎 → (𝑓‘0) = (𝑎‘0))
4342eqeq1d 2653 . . . . . . . 8 (𝑓 = 𝑎 → ((𝑓‘0) = 𝑂 ↔ (𝑎‘0) = 𝑂))
44 fveq1 6228 . . . . . . . . 9 (𝑓 = 𝑎 → (𝑓‘1) = (𝑎‘1))
4544eqeq1d 2653 . . . . . . . 8 (𝑓 = 𝑎 → ((𝑓‘1) = 𝑋 ↔ (𝑎‘1) = 𝑋))
46 coeq2 5313 . . . . . . . . . . . . 13 (𝑓 = 𝑎 → (𝐺𝑓) = (𝐺𝑎))
4746eqeq2d 2661 . . . . . . . . . . . 12 (𝑓 = 𝑎 → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺𝑎)))
4847anbi1d 741 . . . . . . . . . . 11 (𝑓 = 𝑎 → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃)))
4948riotabidv 6653 . . . . . . . . . 10 (𝑓 = 𝑎 → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃)))
5049fveq1d 6231 . . . . . . . . 9 (𝑓 = 𝑎 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1))
5150eqeq1d 2653 . . . . . . . 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 1439 . . . . . . 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 3340 . . . . . 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 1368 . . . . 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 769 . . . . . . . . 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 769 . . . . . . . . 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 769 . . . . . . . . 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 769 . . . . . . . . 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 769 . . . . . . . . 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 769 . . . . . . . . 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 769 . . . . . . . . 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 762 . . . . . . . . 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 762 . . . . . . . . 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 809 . . . . . . . . 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 1129 . . . . . . . . 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 762 . . . . . . . . . 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 1130 . . . . . . . . . 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 2688 . . . . . . . . 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 31427 . . . . . . . 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 1131 . . . . . . . 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 2685 . . . . . . 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 3061 . . . . . 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 2995 . . . . 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 2662 . . . . . . . . 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 1445 . . . . . . . 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 3081 . . . . . . 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 2655 . . . . . . . . 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 329 . . . . . . . 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 3015 . . . . . . 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 747 . . . . . 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 3340 . . . . 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 1364 . . . 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 6228 . . . . . . . . 9 (𝑓 = → (𝑓‘0) = (‘0))
8584eqeq1d 2653 . . . . . . . 8 (𝑓 = → ((𝑓‘0) = 𝑂 ↔ (‘0) = 𝑂))
86 fveq1 6228 . . . . . . . . 9 (𝑓 = → (𝑓‘1) = (‘1))
8786eqeq1d 2653 . . . . . . . 8 (𝑓 = → ((𝑓‘1) = 𝑋 ↔ (‘1) = 𝑋))
88 coeq2 5313 . . . . . . . . . . . . 13 (𝑓 = → (𝐺𝑓) = (𝐺))
8988eqeq2d 2661 . . . . . . . . . . . 12 (𝑓 = → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺)))
9089anbi1d 741 . . . . . . . . . . 11 (𝑓 = → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
9190riotabidv 6653 . . . . . . . . . 10 (𝑓 = → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
9291fveq1d 6231 . . . . . . . . 9 (𝑓 = → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1))
9392eqeq1d 2653 . . . . . . . 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 1439 . . . . . . 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)))
9594cbvrexv 3202 . . . . . 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 2662 . . . . . . . 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 1445 . . . . . . 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 3081 . . . . . 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, 98syl5bb 272 . . . . 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 3435 . . . 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 224 . . 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 3061 . 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 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  ∃!wreu 2943  ∪ cuni 4468   ∘ ccom 5147  ⟶wf 5922  ‘cfv 5926  ℩crio 6650  (class class class)co 6690  0cc0 9974  1c1 9975  [,]cicc 12216   Cn ccn 21076  𝑛-Locally cnlly 21316  IIcii 22725  PConncpconn 31327  SConncsconn 31328   CovMap ccvm 31363 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-ec 7789  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-cn 21079  df-cnp 21080  df-cmp 21238  df-conn 21263  df-lly 21317  df-nlly 21318  df-tx 21413  df-hmeo 21606  df-xms 22172  df-ms 22173  df-tms 22174  df-ii 22727  df-htpy 22816  df-phtpy 22817  df-phtpc 22838  df-pco 22851  df-pconn 31329  df-sconn 31330  df-cvm 31364 This theorem is referenced by:  cvmlift3lem3  31429  cvmlift3lem4  31430
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