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Theorem cvmliftmo 33935
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 725 . . . . 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 725 . . . . 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 725 . . . . 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 725 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝑂𝑌)
11 simplrl 776 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝑓 ∈ (𝐾 Cn 𝐶))
12 simplrr 777 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝑔 ∈ (𝐾 Cn 𝐶))
13 simprll 778 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝐹𝑓) = 𝐺)
14 simprrl 780 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝐹𝑔) = 𝐺)
1513, 14eqtr4d 2776 . . . . 5 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝐹𝑓) = (𝐹𝑔))
16 simprlr 779 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝑓𝑂) = 𝑃)
17 simprrr 781 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → (𝑔𝑂) = 𝑃)
1816, 17eqtr4d 2776 . . . . 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 33934 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝑓 = 𝑔)
2019ex 414 . . 3 ((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) → ((((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃)) → 𝑓 = 𝑔))
2120ralrimivva 3194 . 2 (𝜑 → ∀𝑓 ∈ (𝐾 Cn 𝐶)∀𝑔 ∈ (𝐾 Cn 𝐶)((((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃)) → 𝑓 = 𝑔))
22 coeq2 5815 . . . . 5 (𝑓 = 𝑔 → (𝐹𝑓) = (𝐹𝑔))
2322eqeq1d 2735 . . . 4 (𝑓 = 𝑔 → ((𝐹𝑓) = 𝐺 ↔ (𝐹𝑔) = 𝐺))
24 fveq1 6842 . . . . 5 (𝑓 = 𝑔 → (𝑓𝑂) = (𝑔𝑂))
2524eqeq1d 2735 . . . 4 (𝑓 = 𝑔 → ((𝑓𝑂) = 𝑃 ↔ (𝑔𝑂) = 𝑃))
2623, 25anbi12d 632 . . 3 (𝑓 = 𝑔 → (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ↔ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃)))
2726rmo4 3689 . 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 397   = wceq 1542  wcel 2107  wral 3061  ∃*wrmo 3351   cuni 4866  ccom 5638  cfv 6497  (class class class)co 7358   Cn ccn 22591  Conncconn 22778  𝑛-Locally cnlly 22832   CovMap ccvm 33906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-map 8770  df-en 8887  df-fin 8890  df-fi 9352  df-rest 17309  df-topgen 17330  df-top 22259  df-topon 22276  df-bases 22312  df-cld 22386  df-nei 22465  df-cn 22594  df-conn 22779  df-nlly 22834  df-hmeo 23122  df-cvm 33907
This theorem is referenced by:  cvmliftlem14  33948  cvmlift2lem13  33966  cvmlift3  33979
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