Theorem cvmliftmoi 35277

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 2730	. . 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 35252	. 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 35276	1 ⊢ (𝜑 → 𝑀 = 𝑁)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408   ∖ cdif 3914   ∩ cin 3916  ∅c0 4299  𝒫 cpw 4566  {csn 4592  ∪ cuni 4874   ↦ cmpt 5191  ^◡ccnv 5640   ↾ cres 5643   “ cima 5644   ∘ ccom 5645  ‘cfv 6514  (class class class)co 7390   ↾_t crest 17390   Cn ccn 23118  Conncconn 23305  𝑛-Locally cnlly 23359  Homeochmeo 23647   CovMap ccvm 35249

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-map 8804  df-en 8922  df-fin 8925  df-fi 9369  df-rest 17392  df-topgen 17413  df-top 22788  df-topon 22805  df-bases 22840  df-cld 22913  df-nei 22992  df-cn 23121  df-conn 23306  df-nlly 23361  df-hmeo 23649  df-cvm 35250

This theorem is referenced by:  cvmliftmo  35278  cvmliftphtlem  35311