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Theorem clwlkclwwlkf1 28954
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 28952 . 2 (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
4 fveq2 6842 . . . . . . . 8 (𝑐 = 𝑥 → (2nd𝑐) = (2nd𝑥))
5 2fveq3 6847 . . . . . . . . 9 (𝑐 = 𝑥 → (♯‘(2nd𝑐)) = (♯‘(2nd𝑥)))
65oveq1d 7372 . . . . . . . 8 (𝑐 = 𝑥 → ((♯‘(2nd𝑐)) − 1) = ((♯‘(2nd𝑥)) − 1))
74, 6oveq12d 7375 . . . . . . 7 (𝑐 = 𝑥 → ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)) = ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)))
8 id 22 . . . . . . 7 (𝑥𝐶𝑥𝐶)
9 ovexd 7392 . . . . . . 7 (𝑥𝐶 → ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) ∈ V)
102, 7, 8, 9fvmptd3 6971 . . . . . 6 (𝑥𝐶 → (𝐹𝑥) = ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)))
11 fveq2 6842 . . . . . . . 8 (𝑐 = 𝑦 → (2nd𝑐) = (2nd𝑦))
12 2fveq3 6847 . . . . . . . . 9 (𝑐 = 𝑦 → (♯‘(2nd𝑐)) = (♯‘(2nd𝑦)))
1312oveq1d 7372 . . . . . . . 8 (𝑐 = 𝑦 → ((♯‘(2nd𝑐)) − 1) = ((♯‘(2nd𝑦)) − 1))
1411, 13oveq12d 7375 . . . . . . 7 (𝑐 = 𝑦 → ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)))
15 id 22 . . . . . . 7 (𝑦𝐶𝑦𝐶)
16 ovexd 7392 . . . . . . 7 (𝑦𝐶 → ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) ∈ V)
172, 14, 15, 16fvmptd3 6971 . . . . . 6 (𝑦𝐶 → (𝐹𝑦) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)))
1810, 17eqeqan12d 2750 . . . . 5 ((𝑥𝐶𝑦𝐶) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))))
1918adantl 482 . . . 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 2736 . . . . . . . . . . . . . . 15 (1st𝑥) = (1st𝑥)
23 eqid 2736 . . . . . . . . . . . . . . 15 (2nd𝑥) = (2nd𝑥)
241, 22, 23clwlkclwwlkflem 28948 . . . . . . . . . . . . . 14 (𝑥𝐶 → ((1st𝑥)(Walks‘𝐺)(2nd𝑥) ∧ ((2nd𝑥)‘0) = ((2nd𝑥)‘(♯‘(1st𝑥))) ∧ (♯‘(1st𝑥)) ∈ ℕ))
25 wlklenvm1 28570 . . . . . . . . . . . . . . . 16 ((1st𝑥)(Walks‘𝐺)(2nd𝑥) → (♯‘(1st𝑥)) = ((♯‘(2nd𝑥)) − 1))
2625eqcomd 2742 . . . . . . . . . . . . . . 15 ((1st𝑥)(Walks‘𝐺)(2nd𝑥) → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
27263ad2ant1 1133 . . . . . . . . . . . . . 14 (((1st𝑥)(Walks‘𝐺)(2nd𝑥) ∧ ((2nd𝑥)‘0) = ((2nd𝑥)‘(♯‘(1st𝑥))) ∧ (♯‘(1st𝑥)) ∈ ℕ) → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
2824, 27syl 17 . . . . . . . . . . . . 13 (𝑥𝐶 → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
2928adantr 481 . . . . . . . . . . . 12 ((𝑥𝐶𝑦𝐶) → ((♯‘(2nd𝑥)) − 1) = (♯‘(1st𝑥)))
3029oveq2d 7373 . . . . . . . . . . 11 ((𝑥𝐶𝑦𝐶) → ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑥) prefix (♯‘(1st𝑥))))
31 eqid 2736 . . . . . . . . . . . . . . 15 (1st𝑦) = (1st𝑦)
32 eqid 2736 . . . . . . . . . . . . . . 15 (2nd𝑦) = (2nd𝑦)
331, 31, 32clwlkclwwlkflem 28948 . . . . . . . . . . . . . 14 (𝑦𝐶 → ((1st𝑦)(Walks‘𝐺)(2nd𝑦) ∧ ((2nd𝑦)‘0) = ((2nd𝑦)‘(♯‘(1st𝑦))) ∧ (♯‘(1st𝑦)) ∈ ℕ))
34 wlklenvm1 28570 . . . . . . . . . . . . . . . 16 ((1st𝑦)(Walks‘𝐺)(2nd𝑦) → (♯‘(1st𝑦)) = ((♯‘(2nd𝑦)) − 1))
3534eqcomd 2742 . . . . . . . . . . . . . . 15 ((1st𝑦)(Walks‘𝐺)(2nd𝑦) → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
36353ad2ant1 1133 . . . . . . . . . . . . . 14 (((1st𝑦)(Walks‘𝐺)(2nd𝑦) ∧ ((2nd𝑦)‘0) = ((2nd𝑦)‘(♯‘(1st𝑦))) ∧ (♯‘(1st𝑦)) ∈ ℕ) → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
3733, 36syl 17 . . . . . . . . . . . . 13 (𝑦𝐶 → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
3837adantl 482 . . . . . . . . . . . 12 ((𝑥𝐶𝑦𝐶) → ((♯‘(2nd𝑦)) − 1) = (♯‘(1st𝑦)))
3938oveq2d 7373 . . . . . . . . . . 11 ((𝑥𝐶𝑦𝐶) → ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) = ((2nd𝑦) prefix (♯‘(1st𝑦))))
4030, 39eqeq12d 2752 . . . . . . . . . 10 ((𝑥𝐶𝑦𝐶) → (((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) ↔ ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦)))))
4140adantl 482 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1)) ↔ ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦)))))
4241biimpa 477 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦))))
4320, 21, 423jca 1128 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → (𝑥𝐶𝑦𝐶 ∧ ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦)))))
441, 22, 23, 31, 32clwlkclwwlkf1lem2 28949 . . . . . . 7 ((𝑥𝐶𝑦𝐶 ∧ ((2nd𝑥) prefix (♯‘(1st𝑥))) = ((2nd𝑦) prefix (♯‘(1st𝑦)))) → ((♯‘(1st𝑥)) = (♯‘(1st𝑦)) ∧ ∀𝑖 ∈ (0..^(♯‘(1st𝑥)))((2nd𝑥)‘𝑖) = ((2nd𝑦)‘𝑖)))
45 simpl 483 . . . . . . 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 28950 . . . . . . 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 483 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → 𝐺 ∈ USPGraph)
50 wlkcpr 28577 . . . . . . . . . . . . . 14 (𝑥 ∈ (Walks‘𝐺) ↔ (1st𝑥)(Walks‘𝐺)(2nd𝑥))
5150biimpri 227 . . . . . . . . . . . . 13 ((1st𝑥)(Walks‘𝐺)(2nd𝑥) → 𝑥 ∈ (Walks‘𝐺))
52513ad2ant1 1133 . . . . . . . . . . . 12 (((1st𝑥)(Walks‘𝐺)(2nd𝑥) ∧ ((2nd𝑥)‘0) = ((2nd𝑥)‘(♯‘(1st𝑥))) ∧ (♯‘(1st𝑥)) ∈ ℕ) → 𝑥 ∈ (Walks‘𝐺))
5324, 52syl 17 . . . . . . . . . . 11 (𝑥𝐶𝑥 ∈ (Walks‘𝐺))
54 wlkcpr 28577 . . . . . . . . . . . . . 14 (𝑦 ∈ (Walks‘𝐺) ↔ (1st𝑦)(Walks‘𝐺)(2nd𝑦))
5554biimpri 227 . . . . . . . . . . . . 13 ((1st𝑦)(Walks‘𝐺)(2nd𝑦) → 𝑦 ∈ (Walks‘𝐺))
56553ad2ant1 1133 . . . . . . . . . . . 12 (((1st𝑦)(Walks‘𝐺)(2nd𝑦) ∧ ((2nd𝑦)‘0) = ((2nd𝑦)‘(♯‘(1st𝑦))) ∧ (♯‘(1st𝑦)) ∈ ℕ) → 𝑦 ∈ (Walks‘𝐺))
5733, 56syl 17 . . . . . . . . . . 11 (𝑦𝐶𝑦 ∈ (Walks‘𝐺))
5853, 57anim12i 613 . . . . . . . . . 10 ((𝑥𝐶𝑦𝐶) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
5958adantl 482 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
60 eqidd 2737 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (♯‘(1st𝑥)) = (♯‘(1st𝑥)))
6149, 59, 603jca 1128 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st𝑥)) = (♯‘(1st𝑥))))
6261adantr 481 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ (𝑥𝐶𝑦𝐶)) ∧ ((2nd𝑥) prefix ((♯‘(2nd𝑥)) − 1)) = ((2nd𝑦) prefix ((♯‘(2nd𝑦)) − 1))) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st𝑥)) = (♯‘(1st𝑥))))
63 uspgr2wlkeq 28594 . . . . . . 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 413 . . . 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 3197 . 2 (𝐺 ∈ USPGraph → ∀𝑥𝐶𝑦𝐶 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
69 dff13 7202 . 2 (𝐹:𝐶1-1→(ClWWalks‘𝐺) ↔ (𝐹:𝐶⟶(ClWWalks‘𝐺) ∧ ∀𝑥𝐶𝑦𝐶 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
703, 68, 69sylanbrc 583 1 (𝐺 ∈ USPGraph → 𝐹:𝐶1-1→(ClWWalks‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445   class class class wbr 5105  cmpt 5188  wf 6492  1-1wf1 6493  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  0cc0 11051  1c1 11052  cle 11190  cmin 11385  cn 12153  ...cfz 13424  ..^cfzo 13567  chash 14230   prefix cpfx 14558  USPGraphcuspgr 28099  Walkscwlks 28544  ClWalkscclwlks 28718  ClWWalkscclwwlk 28925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-lsw 14451  df-substr 14529  df-pfx 14559  df-edg 27999  df-uhgr 28009  df-upgr 28033  df-uspgr 28101  df-wlks 28547  df-clwlks 28719  df-clwwlk 28926
This theorem is referenced by:  clwlkclwwlkf1o  28955
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