Theorem cvmliftmo 35647

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 738	. . . . 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 738	. . . . 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 738	. . . . 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 738	. . . . 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 788	. . . . 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 789	. . . . 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 790	. . . . . 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 792	. . . . . 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 2803	. . . . 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 791	. . . . . 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 793	. . . . . 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 2803	. . . . 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 35646	. . . 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 417	. . 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 3208	. 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 5835	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝑔))
23	22	eqeq1d 2767	. . . 4 ⊢ (𝑓 = 𝑔 → ((𝐹 ∘ 𝑓) = 𝐺 ↔ (𝐹 ∘ 𝑔) = 𝐺))
24		fveq1 6870	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑂) = (𝑔‘𝑂))
25	24	eqeq1d 2767	. . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑂) = 𝑃 ↔ (𝑔‘𝑂) = 𝑃))
26	23, 25	anbi12d 643	. . 3 ⊢ (𝑓 = 𝑔 → (((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = 𝐺 ∧ (𝑔‘𝑂) = 𝑃)))
27	26	rmo4 3696	. 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 237	1 ⊢ (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃*wrmo 3369  ∪ cuni 4868   ∘ ccom 5656  ‘cfv 6525  (class class class)co 7400   Cn ccn 23342  Conncconn 23529  𝑛-Locally cnlly 23583   CovMap ccvm 35618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-map 8814  df-en 8932  df-fin 8935  df-fi 9359  df-rest 17465  df-topgen 17486  df-top 23012  df-topon 23029  df-bases 23064  df-cld 23137  df-nei 23216  df-cn 23345  df-conn 23530  df-nlly 23585  df-hmeo 23873  df-cvm 35619

This theorem is referenced by:  cvmliftlem14  35660  cvmlift2lem13  35678  cvmlift3  35691