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Theorem clwlkclwwlkf1 27398
 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 27396 . 2 (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
4 fveq2 6446 . . . . . . . 8 (𝑐 = 𝑥 → (2nd𝑐) = (2nd𝑥))
5 2fveq3 6451 . . . . . . . . 9 (𝑐 = 𝑥 → (♯‘(2nd𝑐)) = (♯‘(2nd𝑥)))
65oveq1d 6937 . . . . . . . 8 (𝑐 = 𝑥 → ((♯‘(2nd𝑐)) − 1) = ((♯‘(2nd𝑥)) − 1))
74, 6oveq12d 6940 . . . . . . 7 (𝑐 = 𝑥 → ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)) = ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)))
8 id 22 . . . . . . 7 (𝑥𝐶𝑥𝐶)
9 ovexd 6956 . . . . . . 7 (𝑥𝐶 → ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) ∈ V)
102, 7, 8, 9fvmptd3 6564 . . . . . 6 (𝑥𝐶 → (𝐹𝑥) = ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)))
11 fveq2 6446 . . . . . . . 8 (𝑐 = 𝑦 → (2nd𝑐) = (2nd𝑦))
12 2fveq3 6451 . . . . . . . . 9 (𝑐 = 𝑦 → (♯‘(2nd𝑐)) = (♯‘(2nd𝑦)))
1312oveq1d 6937 . . . . . . . 8 (𝑐 = 𝑦 → ((♯‘(2nd𝑐)) − 1) = ((♯‘(2nd𝑦)) − 1))
1411, 13oveq12d 6940 . . . . . . 7 (𝑐 = 𝑦 → ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)))
15 id 22 . . . . . . 7 (𝑦𝐶𝑦𝐶)
16 ovexd 6956 . . . . . . 7 (𝑦𝐶 → ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) ∈ V)
172, 14, 15, 16fvmptd3 6564 . . . . . 6 (𝑦𝐶 → (𝐹𝑦) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)))
1810, 17eqeqan12d 2794 . . . . 5 ((𝑥𝐶𝑦𝐶) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))))
1918adantl 475 . . . 4 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))))
20 simplrl 767 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → 𝑥𝐶)
21 simplrr 768 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → 𝑦𝐶)
22 eqid 2778 . . . . . . . . . . . . . . 15 (1st𝑥) = (1st𝑥)
23 eqid 2778 . . . . . . . . . . . . . . 15 (2nd𝑥) = (2nd𝑥)
241, 22, 23clwlkclwwlkflem 27386 . . . . . . . . . . . . . 14 (𝑥𝐶 → ((1st𝑥)(Walks‘𝐺)(2nd𝑥) ∧ ((2nd𝑥)‘0) = ((2nd𝑥)‘(♯‘(1st𝑥))) ∧ (♯‘(1st𝑥)) ∈ ℕ))
25 wlklenvm1 26969 . . . . . . . . . . . . . . . 16 ((1st𝑥)(Walks‘𝐺)(2nd𝑥) → (♯‘(1st𝑥)) = ((♯‘(2nd𝑥)) − 1))
2625eqcomd 2784 . . . . . . . . . . . . . . 15 ((1st𝑥)(Walks‘𝐺)(2nd𝑥) → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
27263ad2ant1 1124 . . . . . . . . . . . . . 14 (((1st𝑥)(Walks‘𝐺)(2nd𝑥) ∧ ((2nd𝑥)‘0) = ((2nd𝑥)‘(♯‘(1st𝑥))) ∧ (♯‘(1st𝑥)) ∈ ℕ) → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
2824, 27syl 17 . . . . . . . . . . . . 13 (𝑥𝐶 → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
2928adantr 474 . . . . . . . . . . . 12 ((𝑥𝐶𝑦𝐶) → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
3029oveq2d 6938 . . . . . . . . . . 11 ((𝑥𝐶𝑦𝐶) → ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑥) prefix (♯‘(1st𝑥))))
31 eqid 2778 . . . . . . . . . . . . . . 15 (1st𝑦) = (1st𝑦)
32 eqid 2778 . . . . . . . . . . . . . . 15 (2nd𝑦) = (2nd𝑦)
331, 31, 32clwlkclwwlkflem 27386 . . . . . . . . . . . . . 14 (𝑦𝐶 → ((1st𝑦)(Walks‘𝐺)(2nd𝑦) ∧ ((2nd𝑦)‘0) = ((2nd𝑦)‘(♯‘(1st𝑦))) ∧ (♯‘(1st𝑦)) ∈ ℕ))
34 wlklenvm1 26969 . . . . . . . . . . . . . . . 16 ((1st𝑦)(Walks‘𝐺)(2nd𝑦) → (♯‘(1st𝑦)) = ((♯‘(2nd𝑦)) − 1))
3534eqcomd 2784 . . . . . . . . . . . . . . 15 ((1st𝑦)(Walks‘𝐺)(2nd𝑦) → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
36353ad2ant1 1124 . . . . . . . . . . . . . 14 (((1st𝑦)(Walks‘𝐺)(2nd𝑦) ∧ ((2nd𝑦)‘0) = ((2nd𝑦)‘(♯‘(1st𝑦))) ∧ (♯‘(1st𝑦)) ∈ ℕ) → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
3733, 36syl 17 . . . . . . . . . . . . 13 (𝑦𝐶 → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
3837adantl 475 . . . . . . . . . . . 12 ((𝑥𝐶𝑦𝐶) → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
3938oveq2d 6938 . . . . . . . . . . 11 ((𝑥𝐶𝑦𝐶) → ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) = ((2nd𝑦) prefix (♯‘(1st𝑦))))
4030, 39eqeq12d 2793 . . . . . . . . . 10 ((𝑥𝐶𝑦𝐶) → (((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) ↔ ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦)))))
4140adantl 475 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) ↔ ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦)))))
4241biimpa 470 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦))))
4320, 21, 423jca 1119 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → (𝑥𝐶𝑦𝐶 ∧ ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦)))))
441, 22, 23, 31, 32clwlkclwwlkf1lem2 27387 . . . . . . 7 ((𝑥𝐶𝑦𝐶 ∧ ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦)))) → ((♯‘(1st𝑥)) = (♯‘(1st𝑦)) ∧ ∀𝑖 ∈ (0..^(♯‘(1st𝑥)))((2nd𝑥)‘𝑖) = ((2nd𝑦)‘𝑖)))
45 simpl 476 . . . . . . 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 27389 . . . . . . 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 476 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → 𝐺 ∈ USPGraph)
50 wlkcpr 26976 . . . . . . . . . . . . . 14 (𝑥 ∈ (Walks‘𝐺) ↔ (1st𝑥)(Walks‘𝐺)(2nd𝑥))
5150biimpri 220 . . . . . . . . . . . . 13 ((1st𝑥)(Walks‘𝐺)(2nd𝑥) → 𝑥 ∈ (Walks‘𝐺))
52513ad2ant1 1124 . . . . . . . . . . . 12 (((1st𝑥)(Walks‘𝐺)(2nd𝑥) ∧ ((2nd𝑥)‘0) = ((2nd𝑥)‘(♯‘(1st𝑥))) ∧ (♯‘(1st𝑥)) ∈ ℕ) → 𝑥 ∈ (Walks‘𝐺))
5324, 52syl 17 . . . . . . . . . . 11 (𝑥𝐶𝑥 ∈ (Walks‘𝐺))
54 wlkcpr 26976 . . . . . . . . . . . . . 14 (𝑦 ∈ (Walks‘𝐺) ↔ (1st𝑦)(Walks‘𝐺)(2nd𝑦))
5554biimpri 220 . . . . . . . . . . . . 13 ((1st𝑦)(Walks‘𝐺)(2nd𝑦) → 𝑦 ∈ (Walks‘𝐺))
56553ad2ant1 1124 . . . . . . . . . . . 12 (((1st𝑦)(Walks‘𝐺)(2nd𝑦) ∧ ((2nd𝑦)‘0) = ((2nd𝑦)‘(♯‘(1st𝑦))) ∧ (♯‘(1st𝑦)) ∈ ℕ) → 𝑦 ∈ (Walks‘𝐺))
5733, 56syl 17 . . . . . . . . . . 11 (𝑦𝐶𝑦 ∈ (Walks‘𝐺))
5853, 57anim12i 606 . . . . . . . . . 10 ((𝑥𝐶𝑦𝐶) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
5958adantl 475 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
60 eqidd 2779 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (♯‘(1st𝑥)) = (♯‘(1st𝑥)))
6149, 59, 603jca 1119 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st𝑥)) = (♯‘(1st𝑥))))
6261adantr 474 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st𝑥)) = (♯‘(1st𝑥))))
63 uspgr2wlkeq 26993 . . . . . . 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 703 . . . . 5 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → 𝑥 = 𝑦)
6665ex 403 . . . 4 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) → 𝑥 = 𝑦))
6719, 66sylbid 232 . . 3 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6867ralrimivva 3153 . 2 (𝐺 ∈ USPGraph → ∀𝑥𝐶𝑦𝐶 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
69 dff13 6784 . 2 (𝐹:𝐶1-1→(ClWWalks‘𝐺) ↔ (𝐹:𝐶⟶(ClWWalks‘𝐺) ∧ ∀𝑥𝐶𝑦𝐶 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
703, 68, 69sylanbrc 578 1 (𝐺 ∈ USPGraph → 𝐹:𝐶1-1→(ClWWalks‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090  {crab 3094  Vcvv 3398   class class class wbr 4886   ↦ cmpt 4965  ⟶wf 6131  –1-1→wf1 6132  ‘cfv 6135  (class class class)co 6922  1st c1st 7443  2nd c2nd 7444  0cc0 10272  1c1 10273   ≤ cle 10412   − cmin 10606  ℕcn 11374  ...cfz 12643  ..^cfzo 12784  ♯chash 13435   prefix cpfx 13779  USPGraphcuspgr 26497  Walkscwlks 26944  ClWalkscclwlks 27122  ClWWalkscclwwlk 27361 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ifp 1047  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-n0 11643  df-xnn0 11715  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-hash 13436  df-word 13600  df-lsw 13653  df-substr 13731  df-pfx 13780  df-edg 26396  df-uhgr 26406  df-upgr 26430  df-uspgr 26499  df-wlks 26947  df-clwlks 27123  df-clwwlk 27362 This theorem is referenced by:  clwlkclwwlkf1o  27399
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