Theorem cvmliftmo 32644

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 725	. . . . 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 725	. . . . 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 725	. . . . 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 725	. . . . 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 2836	. . . . 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 2836	. . . . 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 32643	. . . 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 416	. . 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 3156	. 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 5693	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝑔))
23	22	eqeq1d 2800	. . . 4 ⊢ (𝑓 = 𝑔 → ((𝐹 ∘ 𝑓) = 𝐺 ↔ (𝐹 ∘ 𝑔) = 𝐺))
24		fveq1 6644	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑂) = (𝑔‘𝑂))
25	24	eqeq1d 2800	. . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑂) = 𝑃 ↔ (𝑔‘𝑂) = 𝑃))
26	23, 25	anbi12d 633	. . 3 ⊢ (𝑓 = 𝑔 → (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃)))
27	26	rmo4 3669	. 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 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃*wrmo 3109  ∪ cuni 4800   ∘ ccom 5523  ‘cfv 6324  (class class class)co 7135   Cn ccn 21829  Conncconn 22016  𝑛-Locally cnlly 22070   CovMap ccvm 32615

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-fin 8496  df-fi 8859  df-rest 16688  df-topgen 16709  df-top 21499  df-topon 21516  df-bases 21551  df-cld 21624  df-nei 21703  df-cn 21832  df-conn 22017  df-nlly 22072  df-hmeo 22360  df-cvm 32616

This theorem is referenced by:  cvmliftlem14  32657  cvmlift2lem13  32675  cvmlift3  32688