Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  1wlkpwwlkf1ouspgr Structured version   Visualization version   GIF version

Theorem 1wlkpwwlkf1ouspgr 41076
 Description: The mapping of (ordinary) walks to their sequences of vertices is a bijection in a simple pseudograph. (Contributed by AV, 6-May-2021.)
Hypothesis
Ref Expression
1wlkpwwlkf1ouspgr.f 𝐹 = (𝑤 ∈ (1Walks‘𝐺) ↦ (2nd𝑤))
Assertion
Ref Expression
1wlkpwwlkf1ouspgr (𝐺 ∈ USPGraph → 𝐹:(1Walks‘𝐺)–1-1-onto→(WWalkS‘𝐺))
Distinct variable group:   𝑤,𝐺
Allowed substitution hint:   𝐹(𝑤)

Proof of Theorem 1wlkpwwlkf1ouspgr
Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6113 . . . . . 6 (1st𝑤) ∈ V
2 breq1 4586 . . . . . 6 (𝑓 = (1st𝑤) → (𝑓(1Walks‘𝐺)(2nd𝑤) ↔ (1st𝑤)(1Walks‘𝐺)(2nd𝑤)))
31, 2spcev 3273 . . . . 5 ((1st𝑤)(1Walks‘𝐺)(2nd𝑤) → ∃𝑓 𝑓(1Walks‘𝐺)(2nd𝑤))
4 1wlkiswwlks 41073 . . . . 5 (𝐺 ∈ USPGraph → (∃𝑓 𝑓(1Walks‘𝐺)(2nd𝑤) ↔ (2nd𝑤) ∈ (WWalkS‘𝐺)))
53, 4syl5ib 233 . . . 4 (𝐺 ∈ USPGraph → ((1st𝑤)(1Walks‘𝐺)(2nd𝑤) → (2nd𝑤) ∈ (WWalkS‘𝐺)))
6 1wlkcpr 40833 . . . . 5 (𝑤 ∈ (1Walks‘𝐺) ↔ (1st𝑤)(1Walks‘𝐺)(2nd𝑤))
76biimpi 205 . . . 4 (𝑤 ∈ (1Walks‘𝐺) → (1st𝑤)(1Walks‘𝐺)(2nd𝑤))
85, 7impel 484 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (1Walks‘𝐺)) → (2nd𝑤) ∈ (WWalkS‘𝐺))
9 1wlkpwwlkf1ouspgr.f . . 3 𝐹 = (𝑤 ∈ (1Walks‘𝐺) ↦ (2nd𝑤))
108, 9fmptd 6292 . 2 (𝐺 ∈ USPGraph → 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺))
11 simpr 476 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) → 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺))
129a1i 11 . . . . . . . . 9 (𝑥 ∈ (1Walks‘𝐺) → 𝐹 = (𝑤 ∈ (1Walks‘𝐺) ↦ (2nd𝑤)))
13 fveq2 6103 . . . . . . . . . 10 (𝑤 = 𝑥 → (2nd𝑤) = (2nd𝑥))
1413adantl 481 . . . . . . . . 9 ((𝑥 ∈ (1Walks‘𝐺) ∧ 𝑤 = 𝑥) → (2nd𝑤) = (2nd𝑥))
15 id 22 . . . . . . . . 9 (𝑥 ∈ (1Walks‘𝐺) → 𝑥 ∈ (1Walks‘𝐺))
16 fvex 6113 . . . . . . . . . 10 (2nd𝑥) ∈ V
1716a1i 11 . . . . . . . . 9 (𝑥 ∈ (1Walks‘𝐺) → (2nd𝑥) ∈ V)
1812, 14, 15, 17fvmptd 6197 . . . . . . . 8 (𝑥 ∈ (1Walks‘𝐺) → (𝐹𝑥) = (2nd𝑥))
199a1i 11 . . . . . . . . 9 (𝑦 ∈ (1Walks‘𝐺) → 𝐹 = (𝑤 ∈ (1Walks‘𝐺) ↦ (2nd𝑤)))
20 fveq2 6103 . . . . . . . . . 10 (𝑤 = 𝑦 → (2nd𝑤) = (2nd𝑦))
2120adantl 481 . . . . . . . . 9 ((𝑦 ∈ (1Walks‘𝐺) ∧ 𝑤 = 𝑦) → (2nd𝑤) = (2nd𝑦))
22 id 22 . . . . . . . . 9 (𝑦 ∈ (1Walks‘𝐺) → 𝑦 ∈ (1Walks‘𝐺))
23 fvex 6113 . . . . . . . . . 10 (2nd𝑦) ∈ V
2423a1i 11 . . . . . . . . 9 (𝑦 ∈ (1Walks‘𝐺) → (2nd𝑦) ∈ V)
2519, 21, 22, 24fvmptd 6197 . . . . . . . 8 (𝑦 ∈ (1Walks‘𝐺) → (𝐹𝑦) = (2nd𝑦))
2618, 25eqeqan12d 2626 . . . . . . 7 ((𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 ∈ (1Walks‘𝐺)) → ((𝐹𝑥) = (𝐹𝑦) ↔ (2nd𝑥) = (2nd𝑦)))
2726adantl 481 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) ∧ (𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 ∈ (1Walks‘𝐺))) → ((𝐹𝑥) = (𝐹𝑦) ↔ (2nd𝑥) = (2nd𝑦)))
28 uspgr2wlkeqi 40856 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 ∈ (1Walks‘𝐺)) ∧ (2nd𝑥) = (2nd𝑦)) → 𝑥 = 𝑦)
2928ad4ant134 1288 . . . . . . 7 ((((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) ∧ (𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 ∈ (1Walks‘𝐺))) ∧ (2nd𝑥) = (2nd𝑦)) → 𝑥 = 𝑦)
3029ex 449 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) ∧ (𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 ∈ (1Walks‘𝐺))) → ((2nd𝑥) = (2nd𝑦) → 𝑥 = 𝑦))
3127, 30sylbid 229 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) ∧ (𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 ∈ (1Walks‘𝐺))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3231ralrimivva 2954 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) → ∀𝑥 ∈ (1Walks‘𝐺)∀𝑦 ∈ (1Walks‘𝐺)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
33 dff13 6416 . . . 4 (𝐹:(1Walks‘𝐺)–1-1→(WWalkS‘𝐺) ↔ (𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺) ∧ ∀𝑥 ∈ (1Walks‘𝐺)∀𝑦 ∈ (1Walks‘𝐺)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3411, 32, 33sylanbrc 695 . . 3 ((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) → 𝐹:(1Walks‘𝐺)–1-1→(WWalkS‘𝐺))
35 1wlkiswwlks 41073 . . . . . . . . . 10 (𝐺 ∈ USPGraph → (∃𝑓 𝑓(1Walks‘𝐺)𝑦𝑦 ∈ (WWalkS‘𝐺)))
3635adantr 480 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) → (∃𝑓 𝑓(1Walks‘𝐺)𝑦𝑦 ∈ (WWalkS‘𝐺)))
37 df-br 4584 . . . . . . . . . . 11 (𝑓(1Walks‘𝐺)𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ (1Walks‘𝐺))
38 vex 3176 . . . . . . . . . . . . . 14 𝑓 ∈ V
39 vex 3176 . . . . . . . . . . . . . 14 𝑦 ∈ V
4038, 39op2nd 7068 . . . . . . . . . . . . 13 (2nd ‘⟨𝑓, 𝑦⟩) = 𝑦
4140eqcomi 2619 . . . . . . . . . . . 12 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)
42 opex 4859 . . . . . . . . . . . . 13 𝑓, 𝑦⟩ ∈ V
43 eleq1 2676 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝑓, 𝑦⟩ → (𝑥 ∈ (1Walks‘𝐺) ↔ ⟨𝑓, 𝑦⟩ ∈ (1Walks‘𝐺)))
44 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑥 = ⟨𝑓, 𝑦⟩ → (2nd𝑥) = (2nd ‘⟨𝑓, 𝑦⟩))
4544eqeq2d 2620 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝑓, 𝑦⟩ → (𝑦 = (2nd𝑥) ↔ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)))
4643, 45anbi12d 743 . . . . . . . . . . . . 13 (𝑥 = ⟨𝑓, 𝑦⟩ → ((𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)) ↔ (⟨𝑓, 𝑦⟩ ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩))))
4742, 46spcev 3273 . . . . . . . . . . . 12 ((⟨𝑓, 𝑦⟩ ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)) → ∃𝑥(𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
4841, 47mpan2 703 . . . . . . . . . . 11 (⟨𝑓, 𝑦⟩ ∈ (1Walks‘𝐺) → ∃𝑥(𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
4937, 48sylbi 206 . . . . . . . . . 10 (𝑓(1Walks‘𝐺)𝑦 → ∃𝑥(𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
5049exlimiv 1845 . . . . . . . . 9 (∃𝑓 𝑓(1Walks‘𝐺)𝑦 → ∃𝑥(𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
5136, 50syl6bir 243 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) → (𝑦 ∈ (WWalkS‘𝐺) → ∃𝑥(𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd𝑥))))
5251imp 444 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) ∧ 𝑦 ∈ (WWalkS‘𝐺)) → ∃𝑥(𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
53 df-rex 2902 . . . . . . 7 (∃𝑥 ∈ (1Walks‘𝐺)𝑦 = (2nd𝑥) ↔ ∃𝑥(𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd𝑥)))
5452, 53sylibr 223 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) ∧ 𝑦 ∈ (WWalkS‘𝐺)) → ∃𝑥 ∈ (1Walks‘𝐺)𝑦 = (2nd𝑥))
5518eqeq2d 2620 . . . . . . 7 (𝑥 ∈ (1Walks‘𝐺) → (𝑦 = (𝐹𝑥) ↔ 𝑦 = (2nd𝑥)))
5655rexbiia 3022 . . . . . 6 (∃𝑥 ∈ (1Walks‘𝐺)𝑦 = (𝐹𝑥) ↔ ∃𝑥 ∈ (1Walks‘𝐺)𝑦 = (2nd𝑥))
5754, 56sylibr 223 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) ∧ 𝑦 ∈ (WWalkS‘𝐺)) → ∃𝑥 ∈ (1Walks‘𝐺)𝑦 = (𝐹𝑥))
5857ralrimiva 2949 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) → ∀𝑦 ∈ (WWalkS‘𝐺)∃𝑥 ∈ (1Walks‘𝐺)𝑦 = (𝐹𝑥))
59 dffo3 6282 . . . 4 (𝐹:(1Walks‘𝐺)–onto→(WWalkS‘𝐺) ↔ (𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺) ∧ ∀𝑦 ∈ (WWalkS‘𝐺)∃𝑥 ∈ (1Walks‘𝐺)𝑦 = (𝐹𝑥)))
6011, 58, 59sylanbrc 695 . . 3 ((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) → 𝐹:(1Walks‘𝐺)–onto→(WWalkS‘𝐺))
61 df-f1o 5811 . . 3 (𝐹:(1Walks‘𝐺)–1-1-onto→(WWalkS‘𝐺) ↔ (𝐹:(1Walks‘𝐺)–1-1→(WWalkS‘𝐺) ∧ 𝐹:(1Walks‘𝐺)–onto→(WWalkS‘𝐺)))
6234, 60, 61sylanbrc 695 . 2 ((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) → 𝐹:(1Walks‘𝐺)–1-1-onto→(WWalkS‘𝐺))
6310, 62mpdan 699 1 (𝐺 ∈ USPGraph → 𝐹:(1Walks‘𝐺)–1-1-onto→(WWalkS‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058   USPGraph cuspgr 40378  1Walksc1wlks 40796  WWalkScwwlks 41028 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-uhgr 25724  df-upgr 25749  df-edga 25793  df-uspgr 40380  df-1wlks 40800  df-wlks 40801  df-wwlks 41033 This theorem is referenced by:  1wlkisowwlkupgr  41077
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