Theorem cvmliftmo 35754

Description: A lift of a continuous function from a connected and locally connected space over a covering map is unique when it exists. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by NM, 17-Jun-2017.)

Hypotheses

Ref	Expression
cvmliftmo.b	⊢ 𝐵 = ∪ 𝐶
cvmliftmo.y	⊢ 𝑌 = ∪ 𝐾
cvmliftmo.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmliftmo.k	⊢ (𝜑 → 𝐾 ∈ Conn)
cvmliftmo.l	⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn)
cvmliftmo.o	⊢ (𝜑 → 𝑂 ∈ 𝑌)
cvmliftmo.g	⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
cvmliftmo.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
cvmliftmo.e	⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))

Assertion

Ref	Expression
cvmliftmo	⊢ (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))

Distinct variable groups:   𝐶,𝑓   𝑓,𝐺   𝑓,𝐾   𝑓,𝑂   𝜑,𝑓   𝑓,𝐹   𝑃,𝑓

Allowed substitution hints:   𝐵(𝑓)   𝐽(𝑓)   𝑌(𝑓)

Proof of Theorem cvmliftmo

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmliftmo.b	. . . . 5 ⊢ 𝐵 = ∪ 𝐶
2		cvmliftmo.y	. . . . 5 ⊢ 𝑌 = ∪ 𝐾
3		cvmliftmo.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
4	3	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
5		cvmliftmo.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Conn)
6	5	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃))) → 𝐾 ∈ Conn)
7		cvmliftmo.l	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn)
8	7	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃))) → 𝐾 ∈ 𝑛-Locally Conn)
9		cvmliftmo.o	. . . . . 6 ⊢ (𝜑 → 𝑂 ∈ 𝑌)
10	9	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃))) → 𝑂 ∈ 𝑌)
11		simplrl 788	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃))) → 𝑓 ∈ (𝐾 Cn 𝐶))
12		simplrr 789	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃))) → 𝑔 ∈ (𝐾 Cn 𝐶))
13		simprll 790	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃))) → (𝐹 ∘ 𝑓) = 𝐺)
14		simprrl 792	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃))) → (𝐹 ∘ 𝑔) = 𝐺)
15	13, 14	eqtr4d 2801	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃))) → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝑔))
16		simprlr 791	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃))) → (𝑓‘𝑂) = 𝑃)
17		simprrr 793	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃))) → (𝑔‘𝑂) = 𝑃)
18	16, 17	eqtr4d 2801	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃))) → (𝑓‘𝑂) = (𝑔‘𝑂))
19	1, 2, 4, 6, 8, 10, 11, 12, 15, 18	cvmliftmoi 35753	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃))) → 𝑓 = 𝑔)
20	19	ex 417	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) → ((((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃)) → 𝑓 = 𝑔))
21	20	ralrimivva 3208	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (𝐾 Cn 𝐶)∀𝑔 ∈ (𝐾 Cn 𝐶)((((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃)) → 𝑓 = 𝑔))
22		coeq2 5846	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝑔))
23	22	eqeq1d 2765	. . . 4 ⊢ (𝑓 = 𝑔 → ((𝐹 ∘ 𝑓) = 𝐺 ↔ (𝐹 ∘ 𝑔) = 𝐺))
24		fveq1 6882	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑂) = (𝑔‘𝑂))
25	24	eqeq1d 2765	. . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑂) = 𝑃 ↔ (𝑔‘𝑂) = 𝑃))
26	23, 25	anbi12d 643	. . 3 ⊢ (𝑓 = 𝑔 → (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃)))
27	26	rmo4 3694	. 2 ⊢ (∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ↔ ∀𝑓 ∈ (𝐾 Cn 𝐶)∀𝑔 ∈ (𝐾 Cn 𝐶)((((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃)) → 𝑓 = 𝑔))
28	21, 27	sylibr 237	1 ⊢ (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃*wrmo 3368  ∪ cuni 4873   ∘ ccom 5667  ‘cfv 6538  (class class class)co 7412   Cn ccn 23362  Conncconn 23549  𝑛-Locally cnlly 23603   CovMap ccvm 35725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-map 8827  df-en 8945  df-fin 8948  df-fi 9372  df-rest 17476  df-topgen 17497  df-top 23032  df-topon 23049  df-bases 23084  df-cld 23157  df-nei 23236  df-cn 23365  df-conn 23550  df-nlly 23605  df-hmeo 23893  df-cvm 35726

This theorem is referenced by:  cvmliftlem14  35767  cvmlift2lem13  35785  cvmlift3  35798