Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmlift3lem2 Structured version   Visualization version   GIF version

Theorem cvmlift3lem2 33914
Description: Lemma for cvmlift2 33910. (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 481 . . . 4 ((𝜑𝑋𝑌) → 𝐾 ∈ SConn)
3 sconnpconn 33821 . . . 4 (𝐾 ∈ SConn → 𝐾 ∈ PConn)
42, 3syl 17 . . 3 ((𝜑𝑋𝑌) → 𝐾 ∈ PConn)
5 cvmlift3.o . . . 4 (𝜑𝑂𝑌)
65adantr 481 . . 3 ((𝜑𝑋𝑌) → 𝑂𝑌)
7 simpr 485 . . 3 ((𝜑𝑋𝑌) → 𝑋𝑌)
8 cvmlift3.y . . . 4 𝑌 = 𝐾
98pconncn 33818 . . 3 ((𝐾 ∈ PConn ∧ 𝑂𝑌𝑋𝑌) → ∃𝑎 ∈ (II Cn 𝐾)((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))
104, 6, 7, 9syl3anc 1371 . 2 ((𝜑𝑋𝑌) → ∃𝑎 ∈ (II Cn 𝐾)((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))
11 cvmlift3.b . . . . . . . . 9 𝐵 = 𝐶
12 eqid 2736 . . . . . . . . 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 724 . . . . . . . . 9 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
15 simprl 769 . . . . . . . . . 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 724 . . . . . . . . . 10 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝐺 ∈ (𝐾 Cn 𝐽))
18 cnco 22617 . . . . . . . . . 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 724 . . . . . . . . 9 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → 𝑃𝐵)
22 simprrl 779 . . . . . . . . . . 11 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝑎‘0) = 𝑂)
2322fveq2d 6846 . . . . . . . . . 10 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐺‘(𝑎‘0)) = (𝐺𝑂))
24 iiuni 24244 . . . . . . . . . . . . 13 (0[,]1) = II
2524, 8cnf 22597 . . . . . . . . . . . 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 13386 . . . . . . . . . . 11 0 ∈ (0[,]1)
28 fvco3 6940 . . . . . . . . . . 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 724 . . . . . . . . . 10 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐹𝑃) = (𝐺𝑂))
3223, 29, 313eqtr4rd 2787 . . . . . . . . 9 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝐹𝑃) = ((𝐺𝑎)‘0))
3311, 12, 14, 19, 21, 32cvmliftiota 33895 . . . . . . . 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 1142 . . . . . . 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 22597 . . . . . . 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 13387 . . . . . 6 1 ∈ (0[,]1)
38 ffvelcdm 7032 . . . . . 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 780 . . . . . 6 (((𝜑𝑋𝑌) ∧ (𝑎 ∈ (II Cn 𝐾) ∧ ((𝑎‘0) = 𝑂 ∧ (𝑎‘1) = 𝑋))) → (𝑎‘1) = 𝑋)
41 eqidd 2737 . . . . . 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 6841 . . . . . . . . 9 (𝑓 = 𝑎 → (𝑓‘0) = (𝑎‘0))
4342eqeq1d 2738 . . . . . . . 8 (𝑓 = 𝑎 → ((𝑓‘0) = 𝑂 ↔ (𝑎‘0) = 𝑂))
44 fveq1 6841 . . . . . . . . 9 (𝑓 = 𝑎 → (𝑓‘1) = (𝑎‘1))
4544eqeq1d 2738 . . . . . . . 8 (𝑓 = 𝑎 → ((𝑓‘1) = 𝑋 ↔ (𝑎‘1) = 𝑋))
46 coeq2 5814 . . . . . . . . . . . . 13 (𝑓 = 𝑎 → (𝐺𝑓) = (𝐺𝑎))
4746eqeq2d 2747 . . . . . . . . . . . 12 (𝑓 = 𝑎 → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺𝑎)))
4847anbi1d 630 . . . . . . . . . . 11 (𝑓 = 𝑎 → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃)))
4948riotabidv 7315 . . . . . . . . . 10 (𝑓 = 𝑎 → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃)))
5049fveq1d 6844 . . . . . . . . 9 (𝑓 = 𝑎 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑎) ∧ (𝑔‘0) = 𝑃))‘1))
5150eqeq1d 2738 . . . . . . . 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 1436 . . . . . . 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 3581 . . . . . 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 1372 . . . . 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 730 . . . . . . . . 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 730 . . . . . . . . 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 730 . . . . . . . . 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 730 . . . . . . . . 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 730 . . . . . . . . 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 730 . . . . . . . . 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 730 . . . . . . . . 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 724 . . . . . . . . 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 724 . . . . . . . . 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 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))) → ∈ (II Cn 𝐾))
66 simprr1 1221 . . . . . . . . 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 724 . . . . . . . . . 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 1222 . . . . . . . . . 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 2779 . . . . . . . . 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 33913 . . . . . . . 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 1223 . . . . . . . 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 2776 . . . . . . 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 3153 . . . . . 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 3143 . . . . 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 2748 . . . . . . . . 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 1442 . . . . . . . 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 3175 . . . . . . 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 2740 . . . . . . . . 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 3174 . . . . . . 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 631 . . . . . 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 3581 . . . . 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 835 . . . 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 6841 . . . . . . . . 9 (𝑓 = → (𝑓‘0) = (‘0))
8584eqeq1d 2738 . . . . . . . 8 (𝑓 = → ((𝑓‘0) = 𝑂 ↔ (‘0) = 𝑂))
86 fveq1 6841 . . . . . . . . 9 (𝑓 = → (𝑓‘1) = (‘1))
8786eqeq1d 2738 . . . . . . . 8 (𝑓 = → ((𝑓‘1) = 𝑋 ↔ (‘1) = 𝑋))
88 coeq2 5814 . . . . . . . . . . . . 13 (𝑓 = → (𝐺𝑓) = (𝐺))
8988eqeq2d 2747 . . . . . . . . . . . 12 (𝑓 = → ((𝐹𝑔) = (𝐺𝑓) ↔ (𝐹𝑔) = (𝐺)))
9089anbi1d 630 . . . . . . . . . . 11 (𝑓 = → (((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
9190riotabidv 7315 . . . . . . . . . 10 (𝑓 = → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃)))
9291fveq1d 6844 . . . . . . . . 9 (𝑓 = → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺) ∧ (𝑔‘0) = 𝑃))‘1))
9392eqeq1d 2738 . . . . . . . 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 1436 . . . . . . 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 3226 . . . . . 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 2748 . . . . . . . 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 1442 . . . . . . 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 3175 . . . . . 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 282 . . . . 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 3691 . . . 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 3153 . 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 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  ∃!wreu 3351   cuni 4865  ccom 5637  wf 6492  cfv 6496  crio 7312  (class class class)co 7357  0cc0 11051  1c1 11052  [,]cicc 13267   Cn ccn 22575  𝑛-Locally cnlly 22816  IIcii 24238  PConncpconn 33813  SConncsconn 33814   CovMap ccvm 33849
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-ec 8650  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-cn 22578  df-cnp 22579  df-cmp 22738  df-conn 22763  df-lly 22817  df-nlly 22818  df-tx 22913  df-hmeo 23106  df-xms 23673  df-ms 23674  df-tms 23675  df-ii 24240  df-htpy 24333  df-phtpy 24334  df-phtpc 24355  df-pco 24368  df-pconn 33815  df-sconn 33816  df-cvm 33850
This theorem is referenced by:  cvmlift3lem3  33915  cvmlift3lem4  33916
  Copyright terms: Public domain W3C validator