Theorem cvmliftmoi 35255

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 2729	. . 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 35230	. 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 35254	1 ⊢ (𝜑 → 𝑀 = 𝑁)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3396   ∖ cdif 3902   ∩ cin 3904  ∅c0 4286  𝒫 cpw 4553  {csn 4579  ∪ cuni 4861   ↦ cmpt 5176  ^◡ccnv 5622   ↾ cres 5625   “ cima 5626   ∘ ccom 5627  ‘cfv 6486  (class class class)co 7353   ↾_t crest 17342   Cn ccn 23127  Conncconn 23314  𝑛-Locally cnlly 23368  Homeochmeo 23656   CovMap ccvm 35227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-map 8762  df-en 8880  df-fin 8883  df-fi 9320  df-rest 17344  df-topgen 17365  df-top 22797  df-topon 22814  df-bases 22849  df-cld 22922  df-nei 23001  df-cn 23130  df-conn 23315  df-nlly 23370  df-hmeo 23658  df-cvm 35228

This theorem is referenced by:  cvmliftmo  35256  cvmliftphtlem  35289