MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clwlkclwwlkf1 Structured version   Visualization version   GIF version

Theorem clwlkclwwlkf1 29732
Description: 𝐹 is a one-to-one function from the nonempty closed walks into the closed walks as words in a simple pseudograph. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 29-Oct-2022.)
Hypotheses
Ref Expression
clwlkclwwlkf.c 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}
clwlkclwwlkf.f 𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
Assertion
Ref Expression
clwlkclwwlkf1 (𝐺 ∈ USPGraph → 𝐹:𝐶1-1→(ClWWalks‘𝐺))
Distinct variable groups:   𝑤,𝐺,𝑐   𝐶,𝑐,𝑤   𝐹,𝑐,𝑤

Proof of Theorem clwlkclwwlkf1
Dummy variables 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clwlkclwwlkf.c . . 3 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}
2 clwlkclwwlkf.f . . 3 𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
31, 2clwlkclwwlkf 29730 . 2 (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
4 fveq2 6881 . . . . . . . 8 (𝑐 = 𝑥 → (2nd𝑐) = (2nd𝑥))
5 2fveq3 6886 . . . . . . . . 9 (𝑐 = 𝑥 → (♯‘(2nd𝑐)) = (♯‘(2nd𝑥)))
65oveq1d 7416 . . . . . . . 8 (𝑐 = 𝑥 → ((♯‘(2nd𝑐)) − 1) = ((♯‘(2nd𝑥)) − 1))
74, 6oveq12d 7419 . . . . . . 7 (𝑐 = 𝑥 → ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)) = ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)))
8 id 22 . . . . . . 7 (𝑥𝐶𝑥𝐶)
9 ovexd 7436 . . . . . . 7 (𝑥𝐶 → ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) ∈ V)
102, 7, 8, 9fvmptd3 7011 . . . . . 6 (𝑥𝐶 → (𝐹𝑥) = ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)))
11 fveq2 6881 . . . . . . . 8 (𝑐 = 𝑦 → (2nd𝑐) = (2nd𝑦))
12 2fveq3 6886 . . . . . . . . 9 (𝑐 = 𝑦 → (♯‘(2nd𝑐)) = (♯‘(2nd𝑦)))
1312oveq1d 7416 . . . . . . . 8 (𝑐 = 𝑦 → ((♯‘(2nd𝑐)) − 1) = ((♯‘(2nd𝑦)) − 1))
1411, 13oveq12d 7419 . . . . . . 7 (𝑐 = 𝑦 → ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)))
15 id 22 . . . . . . 7 (𝑦𝐶𝑦𝐶)
16 ovexd 7436 . . . . . . 7 (𝑦𝐶 → ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) ∈ V)
172, 14, 15, 16fvmptd3 7011 . . . . . 6 (𝑦𝐶 → (𝐹𝑦) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)))
1810, 17eqeqan12d 2738 . . . . 5 ((𝑥𝐶𝑦𝐶) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))))
1918adantl 481 . . . 4 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))))
20 simplrl 774 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → 𝑥𝐶)
21 simplrr 775 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → 𝑦𝐶)
22 eqid 2724 . . . . . . . . . . . . . . 15 (1st𝑥) = (1st𝑥)
23 eqid 2724 . . . . . . . . . . . . . . 15 (2nd𝑥) = (2nd𝑥)
241, 22, 23clwlkclwwlkflem 29726 . . . . . . . . . . . . . 14 (𝑥𝐶 → ((1st𝑥)(Walks‘𝐺)(2nd𝑥) ∧ ((2nd𝑥)‘0) = ((2nd𝑥)‘(♯‘(1st𝑥))) ∧ (♯‘(1st𝑥)) ∈ ℕ))
25 wlklenvm1 29348 . . . . . . . . . . . . . . . 16 ((1st𝑥)(Walks‘𝐺)(2nd𝑥) → (♯‘(1st𝑥)) = ((♯‘(2nd𝑥)) − 1))
2625eqcomd 2730 . . . . . . . . . . . . . . 15 ((1st𝑥)(Walks‘𝐺)(2nd𝑥) → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
27263ad2ant1 1130 . . . . . . . . . . . . . 14 (((1st𝑥)(Walks‘𝐺)(2nd𝑥) ∧ ((2nd𝑥)‘0) = ((2nd𝑥)‘(♯‘(1st𝑥))) ∧ (♯‘(1st𝑥)) ∈ ℕ) → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
2824, 27syl 17 . . . . . . . . . . . . 13 (𝑥𝐶 → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
2928adantr 480 . . . . . . . . . . . 12 ((𝑥𝐶𝑦𝐶) → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
3029oveq2d 7417 . . . . . . . . . . 11 ((𝑥𝐶𝑦𝐶) → ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑥) prefix (♯‘(1st𝑥))))
31 eqid 2724 . . . . . . . . . . . . . . 15 (1st𝑦) = (1st𝑦)
32 eqid 2724 . . . . . . . . . . . . . . 15 (2nd𝑦) = (2nd𝑦)
331, 31, 32clwlkclwwlkflem 29726 . . . . . . . . . . . . . 14 (𝑦𝐶 → ((1st𝑦)(Walks‘𝐺)(2nd𝑦) ∧ ((2nd𝑦)‘0) = ((2nd𝑦)‘(♯‘(1st𝑦))) ∧ (♯‘(1st𝑦)) ∈ ℕ))
34 wlklenvm1 29348 . . . . . . . . . . . . . . . 16 ((1st𝑦)(Walks‘𝐺)(2nd𝑦) → (♯‘(1st𝑦)) = ((♯‘(2nd𝑦)) − 1))
3534eqcomd 2730 . . . . . . . . . . . . . . 15 ((1st𝑦)(Walks‘𝐺)(2nd𝑦) → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
36353ad2ant1 1130 . . . . . . . . . . . . . 14 (((1st𝑦)(Walks‘𝐺)(2nd𝑦) ∧ ((2nd𝑦)‘0) = ((2nd𝑦)‘(♯‘(1st𝑦))) ∧ (♯‘(1st𝑦)) ∈ ℕ) → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
3733, 36syl 17 . . . . . . . . . . . . 13 (𝑦𝐶 → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
3837adantl 481 . . . . . . . . . . . 12 ((𝑥𝐶𝑦𝐶) → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
3938oveq2d 7417 . . . . . . . . . . 11 ((𝑥𝐶𝑦𝐶) → ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) = ((2nd𝑦) prefix (♯‘(1st𝑦))))
4030, 39eqeq12d 2740 . . . . . . . . . 10 ((𝑥𝐶𝑦𝐶) → (((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) ↔ ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦)))))
4140adantl 481 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) ↔ ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦)))))
4241biimpa 476 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦))))
4320, 21, 423jca 1125 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → (𝑥𝐶𝑦𝐶 ∧ ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦)))))
441, 22, 23, 31, 32clwlkclwwlkf1lem2 29727 . . . . . . 7 ((𝑥𝐶𝑦𝐶 ∧ ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦)))) → ((♯‘(1st𝑥)) = (♯‘(1st𝑦)) ∧ ∀𝑖 ∈ (0..^(♯‘(1st𝑥)))((2nd𝑥)‘𝑖) = ((2nd𝑦)‘𝑖)))
45 simpl 482 . . . . . . 7 (((♯‘(1st𝑥)) = (♯‘(1st𝑦)) ∧ ∀𝑖 ∈ (0..^(♯‘(1st𝑥)))((2nd𝑥)‘𝑖) = ((2nd𝑦)‘𝑖)) → (♯‘(1st𝑥)) = (♯‘(1st𝑦)))
4643, 44, 453syl 18 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → (♯‘(1st𝑥)) = (♯‘(1st𝑦)))
471, 22, 23, 31, 32clwlkclwwlkf1lem3 29728 . . . . . . 7 ((𝑥𝐶𝑦𝐶 ∧ ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦)))) → ∀𝑖 ∈ (0...(♯‘(1st𝑥)))((2nd𝑥)‘𝑖) = ((2nd𝑦)‘𝑖))
4843, 47syl 17 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → ∀𝑖 ∈ (0...(♯‘(1st𝑥)))((2nd𝑥)‘𝑖) = ((2nd𝑦)‘𝑖))
49 simpl 482 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → 𝐺 ∈ USPGraph)
50 wlkcpr 29355 . . . . . . . . . . . . . 14 (𝑥 ∈ (Walks‘𝐺) ↔ (1st𝑥)(Walks‘𝐺)(2nd𝑥))
5150biimpri 227 . . . . . . . . . . . . 13 ((1st𝑥)(Walks‘𝐺)(2nd𝑥) → 𝑥 ∈ (Walks‘𝐺))
52513ad2ant1 1130 . . . . . . . . . . . 12 (((1st𝑥)(Walks‘𝐺)(2nd𝑥) ∧ ((2nd𝑥)‘0) = ((2nd𝑥)‘(♯‘(1st𝑥))) ∧ (♯‘(1st𝑥)) ∈ ℕ) → 𝑥 ∈ (Walks‘𝐺))
5324, 52syl 17 . . . . . . . . . . 11 (𝑥𝐶𝑥 ∈ (Walks‘𝐺))
54 wlkcpr 29355 . . . . . . . . . . . . . 14 (𝑦 ∈ (Walks‘𝐺) ↔ (1st𝑦)(Walks‘𝐺)(2nd𝑦))
5554biimpri 227 . . . . . . . . . . . . 13 ((1st𝑦)(Walks‘𝐺)(2nd𝑦) → 𝑦 ∈ (Walks‘𝐺))
56553ad2ant1 1130 . . . . . . . . . . . 12 (((1st𝑦)(Walks‘𝐺)(2nd𝑦) ∧ ((2nd𝑦)‘0) = ((2nd𝑦)‘(♯‘(1st𝑦))) ∧ (♯‘(1st𝑦)) ∈ ℕ) → 𝑦 ∈ (Walks‘𝐺))
5733, 56syl 17 . . . . . . . . . . 11 (𝑦𝐶𝑦 ∈ (Walks‘𝐺))
5853, 57anim12i 612 . . . . . . . . . 10 ((𝑥𝐶𝑦𝐶) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
5958adantl 481 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
60 eqidd 2725 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (♯‘(1st𝑥)) = (♯‘(1st𝑥)))
6149, 59, 603jca 1125 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st𝑥)) = (♯‘(1st𝑥))))
6261adantr 480 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st𝑥)) = (♯‘(1st𝑥))))
63 uspgr2wlkeq 29372 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st𝑥)) = (♯‘(1st𝑥))) → (𝑥 = 𝑦 ↔ ((♯‘(1st𝑥)) = (♯‘(1st𝑦)) ∧ ∀𝑖 ∈ (0...(♯‘(1st𝑥)))((2nd𝑥)‘𝑖) = ((2nd𝑦)‘𝑖))))
6462, 63syl 17 . . . . . 6 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → (𝑥 = 𝑦 ↔ ((♯‘(1st𝑥)) = (♯‘(1st𝑦)) ∧ ∀𝑖 ∈ (0...(♯‘(1st𝑥)))((2nd𝑥)‘𝑖) = ((2nd𝑦)‘𝑖))))
6546, 48, 64mpbir2and 710 . . . . 5 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → 𝑥 = 𝑦)
6665ex 412 . . . 4 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) → 𝑥 = 𝑦))
6719, 66sylbid 239 . . 3 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6867ralrimivva 3192 . 2 (𝐺 ∈ USPGraph → ∀𝑥𝐶𝑦𝐶 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
69 dff13 7246 . 2 (𝐹:𝐶1-1→(ClWWalks‘𝐺) ↔ (𝐹:𝐶⟶(ClWWalks‘𝐺) ∧ ∀𝑥𝐶𝑦𝐶 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
703, 68, 69sylanbrc 582 1 (𝐺 ∈ USPGraph → 𝐹:𝐶1-1→(ClWWalks‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3053  {crab 3424  Vcvv 3466   class class class wbr 5138  cmpt 5221  wf 6529  1-1wf1 6530  cfv 6533  (class class class)co 7401  1st c1st 7966  2nd c2nd 7967  0cc0 11106  1c1 11107  cle 11246  cmin 11441  cn 12209  ...cfz 13481  ..^cfzo 13624  chash 14287   prefix cpfx 14617  USPGraphcuspgr 28877  Walkscwlks 29322  ClWalkscclwlks 29496  ClWWalkscclwwlk 29703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1060  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-oadd 8465  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-n0 12470  df-xnn0 12542  df-z 12556  df-uz 12820  df-fz 13482  df-fzo 13625  df-hash 14288  df-word 14462  df-lsw 14510  df-substr 14588  df-pfx 14618  df-edg 28777  df-uhgr 28787  df-upgr 28811  df-uspgr 28879  df-wlks 29325  df-clwlks 29497  df-clwwlk 29704
This theorem is referenced by:  clwlkclwwlkf1o  29733
  Copyright terms: Public domain W3C validator