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Theorem cvmliftmo 32428
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 722 . . . . 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 722 . . . . 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 722 . . . . 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 722 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝑂𝑌)
11 simplrl 773 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝑓 ∈ (𝐾 Cn 𝐶))
12 simplrr 774 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝑔 ∈ (𝐾 Cn 𝐶))
13 simprll 775 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝐹𝑓) = 𝐺)
14 simprrl 777 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝐹𝑔) = 𝐺)
1513, 14eqtr4d 2856 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝐹𝑓) = (𝐹𝑔))
16 simprlr 776 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝑓𝑂) = 𝑃)
17 simprrr 778 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝑔𝑂) = 𝑃)
1816, 17eqtr4d 2856 . . . . 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 32427 . . . 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 3188 . 2 (𝜑 → ∀𝑓 ∈ (𝐾 Cn 𝐶)∀𝑔 ∈ (𝐾 Cn 𝐶)((((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃)) → 𝑓 = 𝑔))
22 coeq2 5722 . . . . 5 (𝑓 = 𝑔 → (𝐹𝑓) = (𝐹𝑔))
2322eqeq1d 2820 . . . 4 (𝑓 = 𝑔 → ((𝐹𝑓) = 𝐺 ↔ (𝐹𝑔) = 𝐺))
24 fveq1 6662 . . . . 5 (𝑓 = 𝑔 → (𝑓𝑂) = (𝑔𝑂))
2524eqeq1d 2820 . . . 4 (𝑓 = 𝑔 → ((𝑓𝑂) = 𝑃 ↔ (𝑔𝑂) = 𝑃))
2623, 25anbi12d 630 . . 3 (𝑓 = 𝑔 → (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ↔ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃)))
2726rmo4 3718 . 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 235 1 (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  ∃*wrmo 3138   cuni 4830  ccom 5552  cfv 6348  (class class class)co 7145   Cn ccn 21760  Conncconn 21947  𝑛-Locally cnlly 22001   CovMap ccvm 32399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-fin 8501  df-fi 8863  df-rest 16684  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-cld 21555  df-nei 21634  df-cn 21763  df-conn 21948  df-nlly 22003  df-hmeo 22291  df-cvm 32400
This theorem is referenced by:  cvmliftlem14  32441  cvmlift2lem13  32459  cvmlift3  32472
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