Theorem cvmliftmo 31266

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 762	. . . . 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 762	. . . . 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 762	. . . . 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 762	. . . . 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 800	. . . . 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 801	. . . . 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 802	. . . . . 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 804	. . . . . 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 2659	. . . . 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 803	. . . . . 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 805	. . . . . 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 2659	. . . . 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 31265	. . . 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 450	. . 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 2971	. 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 5280	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝑔))
23	22	eqeq1d 2624	. . . 4 ⊢ (𝑓 = 𝑔 → ((𝐹 ∘ 𝑓) = 𝐺 ↔ (𝐹 ∘ 𝑔) = 𝐺))
24		fveq1 6190	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑂) = (𝑔‘𝑂))
25	24	eqeq1d 2624	. . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑂) = 𝑃 ↔ (𝑔‘𝑂) = 𝑃))
26	23, 25	anbi12d 747	. . 3 ⊢ (𝑓 = 𝑔 → (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃)))
27	26	rmo4 3399	. 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 224	1 ⊢ (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃*wrmo 2915  ∪ cuni 4436   ∘ ccom 5118  ‘cfv 5888  (class class class)co 6650   Cn ccn 21028  Conncconn 21214  𝑛-Locally cnlly 21268   CovMap ccvm 31237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-nei 20902  df-cn 21031  df-conn 21215  df-nlly 21270  df-hmeo 21558  df-cvm 31238

This theorem is referenced by:  cvmliftlem14  31279  cvmlift2lem13  31297  cvmlift3  31310