Theorem cvmliftmoi 35518

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.)

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)
cvmliftmoi.m	⊢ (𝜑 → 𝑀 ∈ (𝐾 Cn 𝐶))
cvmliftmoi.n	⊢ (𝜑 → 𝑁 ∈ (𝐾 Cn 𝐶))
cvmliftmoi.g	⊢ (𝜑 → (𝐹 ∘ 𝑀) = (𝐹 ∘ 𝑁))
cvmliftmoi.p	⊢ (𝜑 → (𝑀‘𝑂) = (𝑁‘𝑂))

Assertion

Ref	Expression
cvmliftmoi	⊢ (𝜑 → 𝑀 = 𝑁)

Proof of Theorem cvmliftmoi

Dummy variables 𝑏 𝑘 𝑚 𝑟 𝑠 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmliftmo.b	. 2 ⊢ 𝐵 = ∪ 𝐶
2		cvmliftmo.y	. 2 ⊢ 𝑌 = ∪ 𝐾
3		cvmliftmo.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
4		cvmliftmo.k	. 2 ⊢ (𝜑 → 𝐾 ∈ Conn)
5		cvmliftmo.l	. 2 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn)
6		cvmliftmo.o	. 2 ⊢ (𝜑 → 𝑂 ∈ 𝑌)
7		cvmliftmoi.m	. 2 ⊢ (𝜑 → 𝑀 ∈ (𝐾 Cn 𝐶))
8		cvmliftmoi.n	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝐾 Cn 𝐶))
9		cvmliftmoi.g	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝑀) = (𝐹 ∘ 𝑁))
10		cvmliftmoi.p	. 2 ⊢ (𝜑 → (𝑀‘𝑂) = (𝑁‘𝑂))
11		eqid 2740	. . 3 ⊢ (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})
12	11	cvmscbv 35493	. 2 ⊢ (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) = (𝑏 ∈ 𝐽 ↦ {𝑚 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑚 = (◡𝐹 “ 𝑏) ∧ ∀𝑟 ∈ 𝑚 (∀𝑤 ∈ (𝑚 ∖ {𝑟})(𝑟 ∩ 𝑤) = ∅ ∧ (𝐹 ↾ 𝑟) ∈ ((𝐶 ↾t 𝑟)Homeo(𝐽 ↾t 𝑏))))})
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12	cvmliftmolem2 35517	1 ⊢ (𝜑 → 𝑀 = 𝑁)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  {crab 3392   ∖ cdif 3887   ∩ cin 3889  ∅c0 4268  𝒫 cpw 4536  {csn 4562  ∪ cuni 4845   ↦ cmpt 5160  ^◡ccnv 5624   ↾ cres 5627   “ cima 5628   ∘ ccom 5629  ‘cfv 6492  (class class class)co 7363   ↾_t crest 17381   Cn ccn 23214  Conncconn 23401  𝑛-Locally cnlly 23455  Homeochmeo 23743   CovMap ccvm 35490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-map 8772  df-en 8891  df-fin 8894  df-fi 9321  df-rest 17383  df-topgen 17404  df-top 22884  df-topon 22901  df-bases 22936  df-cld 23009  df-nei 23088  df-cn 23217  df-conn 23402  df-nlly 23457  df-hmeo 23745  df-cvm 35491

This theorem is referenced by:  cvmliftmo  35519  cvmliftphtlem  35552