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Theorem cvmlift3lem4 34985
Description: Lemma for cvmlift2 34979. (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))
cvmlift3.h 𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
Assertion
Ref Expression
cvmlift3lem4 ((𝜑𝑋𝑌) → ((𝐻𝑋) = 𝐴 ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴)))
Distinct variable groups:   𝑧,𝑓,𝐴   𝑓,𝑔,𝑧,𝑥   𝑓,𝐽   𝑥,𝑔,𝐽   𝑓,𝐹,𝑔   𝑥,𝑧,𝐹   𝑓,𝐻,𝑔,𝑥,𝑧   𝐵,𝑓,𝑔,𝑥,𝑧   𝑓,𝑋,𝑔,𝑥,𝑧   𝑓,𝐺,𝑔,𝑥,𝑧   𝐶,𝑓,𝑔,𝑥,𝑧   𝜑,𝑓,𝑥   𝑓,𝐾,𝑔,𝑥,𝑧   𝑃,𝑓,𝑔,𝑥,𝑧   𝑓,𝑂,𝑔,𝑥,𝑧   𝑓,𝑌,𝑔,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑔)   𝐴(𝑥,𝑔)   𝐽(𝑧)

Proof of Theorem cvmlift3lem4
StepHypRef Expression
1 cvmlift3.b . . . . 5 𝐵 = 𝐶
2 cvmlift3.y . . . . 5 𝑌 = 𝐾
3 cvmlift3.f . . . . 5 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
4 cvmlift3.k . . . . 5 (𝜑𝐾 ∈ SConn)
5 cvmlift3.l . . . . 5 (𝜑𝐾 ∈ 𝑛-Locally PConn)
6 cvmlift3.o . . . . 5 (𝜑𝑂𝑌)
7 cvmlift3.g . . . . 5 (𝜑𝐺 ∈ (𝐾 Cn 𝐽))
8 cvmlift3.p . . . . 5 (𝜑𝑃𝐵)
9 cvmlift3.e . . . . 5 (𝜑 → (𝐹𝑃) = (𝐺𝑂))
10 cvmlift3.h . . . . 5 𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10cvmlift3lem3 34984 . . . 4 (𝜑𝐻:𝑌𝐵)
1211ffvelcdmda 7087 . . 3 ((𝜑𝑋𝑌) → (𝐻𝑋) ∈ 𝐵)
13 eleq1 2813 . . 3 ((𝐻𝑋) = 𝐴 → ((𝐻𝑋) ∈ 𝐵𝐴𝐵))
1412, 13syl5ibcom 244 . 2 ((𝜑𝑋𝑌) → ((𝐻𝑋) = 𝐴𝐴𝐵))
15 eqid 2725 . . . . . . . . . . 11 (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))
163ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑋𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
17 simprl 769 . . . . . . . . . . . 12 (((𝜑𝑋𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → 𝑓 ∈ (II Cn 𝐾))
187ad2antrr 724 . . . . . . . . . . . 12 (((𝜑𝑋𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → 𝐺 ∈ (𝐾 Cn 𝐽))
19 cnco 23183 . . . . . . . . . . . 12 ((𝑓 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺𝑓) ∈ (II Cn 𝐽))
2017, 18, 19syl2anc 582 . . . . . . . . . . 11 (((𝜑𝑋𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝐺𝑓) ∈ (II Cn 𝐽))
218ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑋𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → 𝑃𝐵)
22 simprr 771 . . . . . . . . . . . . 13 (((𝜑𝑋𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝑓‘0) = 𝑂)
2322fveq2d 6894 . . . . . . . . . . . 12 (((𝜑𝑋𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝐺‘(𝑓‘0)) = (𝐺𝑂))
24 iiuni 24814 . . . . . . . . . . . . . . 15 (0[,]1) = II
2524, 2cnf 23163 . . . . . . . . . . . . . 14 (𝑓 ∈ (II Cn 𝐾) → 𝑓:(0[,]1)⟶𝑌)
2617, 25syl 17 . . . . . . . . . . . . 13 (((𝜑𝑋𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → 𝑓:(0[,]1)⟶𝑌)
27 0elunit 13473 . . . . . . . . . . . . 13 0 ∈ (0[,]1)
28 fvco3 6990 . . . . . . . . . . . . 13 ((𝑓:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺𝑓)‘0) = (𝐺‘(𝑓‘0)))
2926, 27, 28sylancl 584 . . . . . . . . . . . 12 (((𝜑𝑋𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → ((𝐺𝑓)‘0) = (𝐺‘(𝑓‘0)))
309ad2antrr 724 . . . . . . . . . . . 12 (((𝜑𝑋𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝐹𝑃) = (𝐺𝑂))
3123, 29, 303eqtr4rd 2776 . . . . . . . . . . 11 (((𝜑𝑋𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝐹𝑃) = ((𝐺𝑓)‘0))
321, 15, 16, 20, 21, 31cvmliftiota 34964 . . . . . . . . . 10 (((𝜑𝑋𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶) ∧ (𝐹 ∘ (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))) = (𝐺𝑓) ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘0) = 𝑃))
3332simp1d 1139 . . . . . . . . 9 (((𝜑𝑋𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶))
3424, 1cnf 23163 . . . . . . . . 9 ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶) → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵)
3533, 34syl 17 . . . . . . . 8 (((𝜑𝑋𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵)
36 1elunit 13474 . . . . . . . 8 1 ∈ (0[,]1)
37 ffvelcdm 7084 . . . . . . . 8 (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) ∈ 𝐵)
3835, 36, 37sylancl 584 . . . . . . 7 (((𝜑𝑋𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) ∈ 𝐵)
39 eleq1 2813 . . . . . . 7 (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴 → (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) ∈ 𝐵𝐴𝐵))
4038, 39syl5ibcom 244 . . . . . 6 (((𝜑𝑋𝑌) ∧ (𝑓 ∈ (II Cn 𝐾) ∧ (𝑓‘0) = 𝑂)) → (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴𝐴𝐵))
4140expr 455 . . . . 5 (((𝜑𝑋𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) → ((𝑓‘0) = 𝑂 → (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴𝐴𝐵)))
4241a1dd 50 . . . 4 (((𝜑𝑋𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) → ((𝑓‘0) = 𝑂 → ((𝑓‘1) = 𝑋 → (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴𝐴𝐵))))
43423impd 1345 . . 3 (((𝜑𝑋𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴) → 𝐴𝐵))
4443rexlimdva 3145 . 2 ((𝜑𝑋𝑌) → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴) → 𝐴𝐵))
45 eqeq2 2737 . . . . . . . . . . 11 (𝑥 = 𝑋 → ((𝑓‘1) = 𝑥 ↔ (𝑓‘1) = 𝑋))
46453anbi2d 1437 . . . . . . . . . 10 (𝑥 = 𝑋 → (((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) = 𝑧)))
4746rexbidv 3169 . . . . . . . . 9 (𝑥 = 𝑋 → (∃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) = 𝑧)))
4847riotabidv 7371 . . . . . . . 8 (𝑥 = 𝑋 → (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)) = (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
49 riotaex 7373 . . . . . . . 8 (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)) ∈ V
5048, 10, 49fvmpt 6998 . . . . . . 7 (𝑋𝑌 → (𝐻𝑋) = (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
5150adantl 480 . . . . . 6 ((𝜑𝑋𝑌) → (𝐻𝑋) = (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
5251eqeq1d 2727 . . . . 5 ((𝜑𝑋𝑌) → ((𝐻𝑋) = 𝐴 ↔ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)) = 𝐴))
5352adantl 480 . . . 4 ((𝐴𝐵 ∧ (𝜑𝑋𝑌)) → ((𝐻𝑋) = 𝐴 ↔ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)) = 𝐴))
541, 2, 3, 4, 5, 6, 7, 8, 9cvmlift3lem2 34983 . . . . 5 ((𝜑𝑋𝑌) → ∃!𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))
55 eqeq2 2737 . . . . . . . 8 (𝑧 = 𝐴 → (((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧 ↔ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴))
56553anbi3d 1438 . . . . . . 7 (𝑧 = 𝐴 → (((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) = 𝐴)))
5756rexbidv 3169 . . . . . 6 (𝑧 = 𝐴 → (∃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) = 𝐴)))
5857riota2 7395 . . . . 5 ((𝐴𝐵 ∧ ∃!𝑧𝐵𝑓 ∈ (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) = 𝐴) ↔ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)) = 𝐴))
5954, 58sylan2 591 . . . 4 ((𝐴𝐵 ∧ (𝜑𝑋𝑌)) → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴) ↔ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)) = 𝐴))
6053, 59bitr4d 281 . . 3 ((𝐴𝐵 ∧ (𝜑𝑋𝑌)) → ((𝐻𝑋) = 𝐴 ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴)))
6160expcom 412 . 2 ((𝜑𝑋𝑌) → (𝐴𝐵 → ((𝐻𝑋) = 𝐴 ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴))))
6214, 44, 61pm5.21ndd 378 1 ((𝜑𝑋𝑌) → ((𝐻𝑋) = 𝐴 ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wrex 3060  ∃!wreu 3362   cuni 4904  cmpt 5227  ccom 5677  wf 6539  cfv 6543  crio 7368  (class class class)co 7413  0cc0 11133  1c1 11134  [,]cicc 13354   Cn ccn 23141  𝑛-Locally cnlly 23382  IIcii 24808  PConncpconn 34882  SConncsconn 34883   CovMap ccvm 34918
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 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-inf2 9659  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210  ax-pre-sup 11211  ax-addf 11212
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-iin 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7679  df-om 7866  df-1st 7987  df-2nd 7988  df-supp 8159  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8718  df-ec 8720  df-map 8840  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9381  df-fi 9429  df-sup 9460  df-inf 9461  df-oi 9528  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-div 11897  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-9 12307  df-n0 12498  df-z 12584  df-dec 12703  df-uz 12848  df-q 12958  df-rp 13002  df-xneg 13119  df-xadd 13120  df-xmul 13121  df-ioo 13355  df-ico 13357  df-icc 13358  df-fz 13512  df-fzo 13655  df-fl 13784  df-seq 13994  df-exp 14054  df-hash 14317  df-cj 15073  df-re 15074  df-im 15075  df-sqrt 15209  df-abs 15210  df-clim 15459  df-sum 15660  df-struct 17110  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-ress 17204  df-plusg 17240  df-mulr 17241  df-starv 17242  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-unif 17250  df-hom 17251  df-cco 17252  df-rest 17398  df-topn 17399  df-0g 17417  df-gsum 17418  df-topgen 17419  df-pt 17420  df-prds 17423  df-xrs 17478  df-qtop 17483  df-imas 17484  df-xps 17486  df-mre 17560  df-mrc 17561  df-acs 17563  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-submnd 18735  df-mulg 19023  df-cntz 19267  df-cmn 19736  df-psmet 21270  df-xmet 21271  df-met 21272  df-bl 21273  df-mopn 21274  df-cnfld 21279  df-top 22809  df-topon 22826  df-topsp 22848  df-bases 22862  df-cld 22936  df-ntr 22937  df-cls 22938  df-nei 23015  df-cn 23144  df-cnp 23145  df-cmp 23304  df-conn 23329  df-lly 23383  df-nlly 23384  df-tx 23479  df-hmeo 23672  df-xms 24239  df-ms 24240  df-tms 24241  df-ii 24810  df-cncf 24811  df-htpy 24909  df-phtpy 24910  df-phtpc 24931  df-pco 24945  df-pconn 34884  df-sconn 34885  df-cvm 34919
This theorem is referenced by:  cvmlift3lem5  34986  cvmlift3lem6  34987  cvmlift3lem7  34988  cvmlift3lem9  34990
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