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Theorem cvmliftmo 32538
 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 2859 . . . . 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 2859 . . . . 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 32537 . . . 4 (((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) ∧ (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃))) → 𝑓 = 𝑔)
2019ex 416 . . 3 ((𝜑 ∧ (𝑓 ∈ (𝐾 Cn 𝐶) ∧ 𝑔 ∈ (𝐾 Cn 𝐶))) → ((((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃)) → 𝑓 = 𝑔))
2120ralrimivva 3179 . 2 (𝜑 → ∀𝑓 ∈ (𝐾 Cn 𝐶)∀𝑔 ∈ (𝐾 Cn 𝐶)((((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ∧ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃)) → 𝑓 = 𝑔))
22 coeq2 5702 . . . . 5 (𝑓 = 𝑔 → (𝐹𝑓) = (𝐹𝑔))
2322eqeq1d 2823 . . . 4 (𝑓 = 𝑔 → ((𝐹𝑓) = 𝐺 ↔ (𝐹𝑔) = 𝐺))
24 fveq1 6642 . . . . 5 (𝑓 = 𝑔 → (𝑓𝑂) = (𝑔𝑂))
2524eqeq1d 2823 . . . 4 (𝑓 = 𝑔 → ((𝑓𝑂) = 𝑃 ↔ (𝑔𝑂) = 𝑃))
2623, 25anbi12d 633 . . 3 (𝑓 = 𝑔 → (((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃) ↔ ((𝐹𝑔) = 𝐺 ∧ (𝑔𝑂) = 𝑃)))
2726rmo4 3698 . 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 237 1 (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3126  ∃*wrmo 3129  ∪ cuni 4811   ∘ ccom 5532  ‘cfv 6328  (class class class)co 7130   Cn ccn 21807  Conncconn 21994  𝑛-Locally cnlly 22048   CovMap ccvm 32509 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-oadd 8081  df-er 8264  df-map 8383  df-en 8485  df-fin 8488  df-fi 8851  df-rest 16674  df-topgen 16695  df-top 21477  df-topon 21494  df-bases 21529  df-cld 21602  df-nei 21681  df-cn 21810  df-conn 21995  df-nlly 22050  df-hmeo 22338  df-cvm 32510 This theorem is referenced by:  cvmliftlem14  32551  cvmlift2lem13  32569  cvmlift3  32582
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