Theorem cvmliftmoi 35597

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 2761	. . 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 35572	. 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 35596	1 ⊢ (𝜑 → 𝑀 = 𝑁)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413   ∖ cdif 3901   ∩ cin 3903  ∅c0 4285  𝒫 cpw 4554  {csn 4581  ∪ cuni 4864   ↦ cmpt 5180  ^◡ccnv 5644   ↾ cres 5647   “ cima 5648   ∘ ccom 5649  ‘cfv 6517  (class class class)co 7392   ↾_t crest 17432   Cn ccn 23264  Conncconn 23451  𝑛-Locally cnlly 23505  Homeochmeo 23793   CovMap ccvm 35569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-map 8805  df-en 8924  df-fin 8927  df-fi 9354  df-rest 17434  df-topgen 17455  df-top 22934  df-topon 22951  df-bases 22986  df-cld 23059  df-nei 23138  df-cn 23267  df-conn 23452  df-nlly 23507  df-hmeo 23795  df-cvm 35570

This theorem is referenced by:  cvmliftmo  35598  cvmliftphtlem  35631