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Theorem cvmliftmo 34344
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 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
cvmliftmo.k (𝜑𝐾 ∈ Conn)
cvmliftmo.l (𝜑𝐾 ∈ 𝑛-Locally Conn)
cvmliftmo.o (𝜑𝑂𝑌)
cvmliftmo.g (𝜑𝐺 ∈ (𝐾 Cn 𝐽))
cvmliftmo.p (𝜑𝑃𝐵)
cvmliftmo.e (𝜑 → (𝐹𝑃) = (𝐺𝑂))
Assertion
Ref Expression
cvmliftmo (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 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.
StepHypRef Expression
1 cvmliftmo.b . . . . 5 𝐵 = 𝐶
2 cvmliftmo.y . . . . 5 𝑌 = 𝐾
3 cvmliftmo.f . . . . . 6 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
43ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝐹 ∈ (𝐶 CovMap 𝐽))
5 cvmliftmo.k . . . . . 6 (𝜑𝐾 ∈ Conn)
65ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝐾 ∈ Conn)
7 cvmliftmo.l . . . . . 6 (𝜑𝐾 ∈ 𝑛-Locally Conn)
87ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝐾 ∈ 𝑛-Locally Conn)
9 cvmliftmo.o . . . . . 6 (𝜑𝑂𝑌)
109ad2antrr 724 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝑂𝑌)
11 simplrl 775 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝑓 ∈ (𝐾 Cn 𝐶))
12 simplrr 776 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝑔 ∈ (𝐾 Cn 𝐶))
13 simprll 777 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝐹𝑓) = 𝐺)
14 simprrl 779 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝐹𝑔) = 𝐺)
1513, 14eqtr4d 2775 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝐹𝑓) = (𝐹𝑔))
16 simprlr 778 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝑓𝑂) = 𝑃)
17 simprrr 780 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝑔𝑂) = 𝑃)
1816, 17eqtr4d 2775 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝑓𝑂) = (𝑔𝑂))
191, 2, 4, 6, 8, 10, 11, 12, 15, 18cvmliftmoi 34343 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝑓 = 𝑔)
2019ex 413 . . 3 ((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) → ((((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃)) → 𝑓 = 𝑔))
2120ralrimivva 3200 . 2 (𝜑 → ∀𝑓 ∈ (𝐾 Cn 𝐶)∀𝑔 ∈ (𝐾 Cn 𝐶)((((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃)) → 𝑓 = 𝑔))
22 coeq2 5858 . . . . 5 (𝑓 = 𝑔 → (𝐹𝑓) = (𝐹𝑔))
2322eqeq1d 2734 . . . 4 (𝑓 = 𝑔 → ((𝐹𝑓) = 𝐺 ↔ (𝐹𝑔) = 𝐺))
24 fveq1 6890 . . . . 5 (𝑓 = 𝑔 → (𝑓𝑂) = (𝑔𝑂))
2524eqeq1d 2734 . . . 4 (𝑓 = 𝑔 → ((𝑓𝑂) = 𝑃 ↔ (𝑔𝑂) = 𝑃))
2623, 25anbi12d 631 . . 3 (𝑓 = 𝑔 → (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ↔ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃)))
2726rmo4 3726 . 2 (∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ↔ ∀𝑓 ∈ (𝐾 Cn 𝐶)∀𝑔 ∈ (𝐾 Cn 𝐶)((((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃)) → 𝑓 = 𝑔))
2821, 27sylibr 233 1 (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061  ∃*wrmo 3375   cuni 4908  ccom 5680  cfv 6543  (class class class)co 7411   Cn ccn 22735  Conncconn 22922  𝑛-Locally cnlly 22976   CovMap ccvm 34315
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  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-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-map 8824  df-en 8942  df-fin 8945  df-fi 9408  df-rest 17370  df-topgen 17391  df-top 22403  df-topon 22420  df-bases 22456  df-cld 22530  df-nei 22609  df-cn 22738  df-conn 22923  df-nlly 22978  df-hmeo 23266  df-cvm 34316
This theorem is referenced by:  cvmliftlem14  34357  cvmlift2lem13  34375  cvmlift3  34388
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