Theorem cvmliftmo 35264

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 726	. . . . 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 726	. . . . 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 726	. . . . 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 726	. . . . 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 776	. . . . 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 777	. . . . 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 778	. . . . . 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 780	. . . . . 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 2772	. . . . 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 779	. . . . . 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 781	. . . . . 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 2772	. . . . 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 35263	. . . 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 412	. . 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 3189	. 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 5849	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝑔))
23	22	eqeq1d 2736	. . . 4 ⊢ (𝑓 = 𝑔 → ((𝐹 ∘ 𝑓) = 𝐺 ↔ (𝐹 ∘ 𝑔) = 𝐺))
24		fveq1 6885	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑂) = (𝑔‘𝑂))
25	24	eqeq1d 2736	. . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑂) = 𝑃 ↔ (𝑔‘𝑂) = 𝑃))
26	23, 25	anbi12d 632	. . 3 ⊢ (𝑓 = 𝑔 → (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃)))
27	26	rmo4 3718	. 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 234	1 ⊢ (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ∃*wrmo 3362  ∪ cuni 4887   ∘ ccom 5669  ‘cfv 6541  (class class class)co 7413   Cn ccn 23179  Conncconn 23366  𝑛-Locally cnlly 23420   CovMap ccvm 35235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-int 4927  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-map 8850  df-en 8968  df-fin 8971  df-fi 9433  df-rest 17439  df-topgen 17460  df-top 22849  df-topon 22866  df-bases 22901  df-cld 22974  df-nei 23053  df-cn 23182  df-conn 23367  df-nlly 23422  df-hmeo 23710  df-cvm 35236

This theorem is referenced by:  cvmliftlem14  35277  cvmlift2lem13  35295  cvmlift3  35308