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Theorem clwlkclwwlkf1 27791
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 27789 . 2 (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
4 fveq2 6673 . . . . . . . 8 (𝑐 = 𝑥 → (2nd𝑐) = (2nd𝑥))
5 2fveq3 6678 . . . . . . . . 9 (𝑐 = 𝑥 → (♯‘(2nd𝑐)) = (♯‘(2nd𝑥)))
65oveq1d 7174 . . . . . . . 8 (𝑐 = 𝑥 → ((♯‘(2nd𝑐)) − 1) = ((♯‘(2nd𝑥)) − 1))
74, 6oveq12d 7177 . . . . . . 7 (𝑐 = 𝑥 → ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)) = ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)))
8 id 22 . . . . . . 7 (𝑥𝐶𝑥𝐶)
9 ovexd 7194 . . . . . . 7 (𝑥𝐶 → ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) ∈ V)
102, 7, 8, 9fvmptd3 6794 . . . . . 6 (𝑥𝐶 → (𝐹𝑥) = ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)))
11 fveq2 6673 . . . . . . . 8 (𝑐 = 𝑦 → (2nd𝑐) = (2nd𝑦))
12 2fveq3 6678 . . . . . . . . 9 (𝑐 = 𝑦 → (♯‘(2nd𝑐)) = (♯‘(2nd𝑦)))
1312oveq1d 7174 . . . . . . . 8 (𝑐 = 𝑦 → ((♯‘(2nd𝑐)) − 1) = ((♯‘(2nd𝑦)) − 1))
1411, 13oveq12d 7177 . . . . . . 7 (𝑐 = 𝑦 → ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)))
15 id 22 . . . . . . 7 (𝑦𝐶𝑦𝐶)
16 ovexd 7194 . . . . . . 7 (𝑦𝐶 → ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) ∈ V)
172, 14, 15, 16fvmptd3 6794 . . . . . 6 (𝑦𝐶 → (𝐹𝑦) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)))
1810, 17eqeqan12d 2841 . . . . 5 ((𝑥𝐶𝑦𝐶) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))))
1918adantl 484 . . . 4 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))))
20 simplrl 775 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → 𝑥𝐶)
21 simplrr 776 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → 𝑦𝐶)
22 eqid 2824 . . . . . . . . . . . . . . 15 (1st𝑥) = (1st𝑥)
23 eqid 2824 . . . . . . . . . . . . . . 15 (2nd𝑥) = (2nd𝑥)
241, 22, 23clwlkclwwlkflem 27785 . . . . . . . . . . . . . 14 (𝑥𝐶 → ((1st𝑥)(Walks‘𝐺)(2nd𝑥) ∧ ((2nd𝑥)‘0) = ((2nd𝑥)‘(♯‘(1st𝑥))) ∧ (♯‘(1st𝑥)) ∈ ℕ))
25 wlklenvm1 27406 . . . . . . . . . . . . . . . 16 ((1st𝑥)(Walks‘𝐺)(2nd𝑥) → (♯‘(1st𝑥)) = ((♯‘(2nd𝑥)) − 1))
2625eqcomd 2830 . . . . . . . . . . . . . . 15 ((1st𝑥)(Walks‘𝐺)(2nd𝑥) → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
27263ad2ant1 1129 . . . . . . . . . . . . . 14 (((1st𝑥)(Walks‘𝐺)(2nd𝑥) ∧ ((2nd𝑥)‘0) = ((2nd𝑥)‘(♯‘(1st𝑥))) ∧ (♯‘(1st𝑥)) ∈ ℕ) → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
2824, 27syl 17 . . . . . . . . . . . . 13 (𝑥𝐶 → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
2928adantr 483 . . . . . . . . . . . 12 ((𝑥𝐶𝑦𝐶) → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
3029oveq2d 7175 . . . . . . . . . . 11 ((𝑥𝐶𝑦𝐶) → ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑥) prefix (♯‘(1st𝑥))))
31 eqid 2824 . . . . . . . . . . . . . . 15 (1st𝑦) = (1st𝑦)
32 eqid 2824 . . . . . . . . . . . . . . 15 (2nd𝑦) = (2nd𝑦)
331, 31, 32clwlkclwwlkflem 27785 . . . . . . . . . . . . . 14 (𝑦𝐶 → ((1st𝑦)(Walks‘𝐺)(2nd𝑦) ∧ ((2nd𝑦)‘0) = ((2nd𝑦)‘(♯‘(1st𝑦))) ∧ (♯‘(1st𝑦)) ∈ ℕ))
34 wlklenvm1 27406 . . . . . . . . . . . . . . . 16 ((1st𝑦)(Walks‘𝐺)(2nd𝑦) → (♯‘(1st𝑦)) = ((♯‘(2nd𝑦)) − 1))
3534eqcomd 2830 . . . . . . . . . . . . . . 15 ((1st𝑦)(Walks‘𝐺)(2nd𝑦) → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
36353ad2ant1 1129 . . . . . . . . . . . . . 14 (((1st𝑦)(Walks‘𝐺)(2nd𝑦) ∧ ((2nd𝑦)‘0) = ((2nd𝑦)‘(♯‘(1st𝑦))) ∧ (♯‘(1st𝑦)) ∈ ℕ) → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
3733, 36syl 17 . . . . . . . . . . . . 13 (𝑦𝐶 → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
3837adantl 484 . . . . . . . . . . . 12 ((𝑥𝐶𝑦𝐶) → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
3938oveq2d 7175 . . . . . . . . . . 11 ((𝑥𝐶𝑦𝐶) → ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) = ((2nd𝑦) prefix (♯‘(1st𝑦))))
4030, 39eqeq12d 2840 . . . . . . . . . 10 ((𝑥𝐶𝑦𝐶) → (((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) ↔ ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦)))))
4140adantl 484 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) ↔ ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦)))))
4241biimpa 479 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦))))
4320, 21, 423jca 1124 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → (𝑥𝐶𝑦𝐶 ∧ ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦)))))
441, 22, 23, 31, 32clwlkclwwlkf1lem2 27786 . . . . . . 7 ((𝑥𝐶𝑦𝐶 ∧ ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦)))) → ((♯‘(1st𝑥)) = (♯‘(1st𝑦)) ∧ ∀𝑖 ∈ (0..^(♯‘(1st𝑥)))((2nd𝑥)‘𝑖) = ((2nd𝑦)‘𝑖)))
45 simpl 485 . . . . . . 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 27787 . . . . . . 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 485 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → 𝐺 ∈ USPGraph)
50 wlkcpr 27413 . . . . . . . . . . . . . 14 (𝑥 ∈ (Walks‘𝐺) ↔ (1st𝑥)(Walks‘𝐺)(2nd𝑥))
5150biimpri 230 . . . . . . . . . . . . 13 ((1st𝑥)(Walks‘𝐺)(2nd𝑥) → 𝑥 ∈ (Walks‘𝐺))
52513ad2ant1 1129 . . . . . . . . . . . 12 (((1st𝑥)(Walks‘𝐺)(2nd𝑥) ∧ ((2nd𝑥)‘0) = ((2nd𝑥)‘(♯‘(1st𝑥))) ∧ (♯‘(1st𝑥)) ∈ ℕ) → 𝑥 ∈ (Walks‘𝐺))
5324, 52syl 17 . . . . . . . . . . 11 (𝑥𝐶𝑥 ∈ (Walks‘𝐺))
54 wlkcpr 27413 . . . . . . . . . . . . . 14 (𝑦 ∈ (Walks‘𝐺) ↔ (1st𝑦)(Walks‘𝐺)(2nd𝑦))
5554biimpri 230 . . . . . . . . . . . . 13 ((1st𝑦)(Walks‘𝐺)(2nd𝑦) → 𝑦 ∈ (Walks‘𝐺))
56553ad2ant1 1129 . . . . . . . . . . . 12 (((1st𝑦)(Walks‘𝐺)(2nd𝑦) ∧ ((2nd𝑦)‘0) = ((2nd𝑦)‘(♯‘(1st𝑦))) ∧ (♯‘(1st𝑦)) ∈ ℕ) → 𝑦 ∈ (Walks‘𝐺))
5733, 56syl 17 . . . . . . . . . . 11 (𝑦𝐶𝑦 ∈ (Walks‘𝐺))
5853, 57anim12i 614 . . . . . . . . . 10 ((𝑥𝐶𝑦𝐶) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
5958adantl 484 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
60 eqidd 2825 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (♯‘(1st𝑥)) = (♯‘(1st𝑥)))
6149, 59, 603jca 1124 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st𝑥)) = (♯‘(1st𝑥))))
6261adantr 483 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st𝑥)) = (♯‘(1st𝑥))))
63 uspgr2wlkeq 27430 . . . . . . 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 711 . . . . 5 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → 𝑥 = 𝑦)
6665ex 415 . . . 4 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) → 𝑥 = 𝑦))
6719, 66sylbid 242 . . 3 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6867ralrimivva 3194 . 2 (𝐺 ∈ USPGraph → ∀𝑥𝐶𝑦𝐶 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
69 dff13 7016 . 2 (𝐹:𝐶1-1→(ClWWalks‘𝐺) ↔ (𝐹:𝐶⟶(ClWWalks‘𝐺) ∧ ∀𝑥𝐶𝑦𝐶 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
703, 68, 69sylanbrc 585 1 (𝐺 ∈ USPGraph → 𝐹:𝐶1-1→(ClWWalks‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1536  wcel 2113  wral 3141  {crab 3145  Vcvv 3497   class class class wbr 5069  cmpt 5149  wf 6354  1-1wf1 6355  cfv 6358  (class class class)co 7159  1st c1st 7690  2nd c2nd 7691  0cc0 10540  1c1 10541  cle 10679  cmin 10873  cn 11641  ...cfz 12895  ..^cfzo 13036  chash 13693   prefix cpfx 14035  USPGraphcuspgr 26936  Walkscwlks 27381  ClWalkscclwlks 27554  ClWWalkscclwwlk 27762
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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-2o 8106  df-oadd 8109  df-er 8292  df-map 8411  df-pm 8412  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-dju 9333  df-card 9371  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-2 11703  df-n0 11901  df-xnn0 11971  df-z 11985  df-uz 12247  df-fz 12896  df-fzo 13037  df-hash 13694  df-word 13865  df-lsw 13918  df-substr 14006  df-pfx 14036  df-edg 26836  df-uhgr 26846  df-upgr 26870  df-uspgr 26938  df-wlks 27384  df-clwlks 27555  df-clwwlk 27763
This theorem is referenced by:  clwlkclwwlkf1o  27792
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