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) ↦ (2nd ‘𝑤))

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.

Step	Hyp	Ref	Expression
1		fvex 6113	. . . . . 6 ⊢ (1st ‘𝑤) ∈ V
2		breq1 4586	. . . . . 6 ⊢ (𝑓 = (1st ‘𝑤) → (𝑓(1Walks‘𝐺)(2nd ‘𝑤) ↔ (1st ‘𝑤)(1Walks‘𝐺)(2nd ‘𝑤)))
3	1, 2	spcev 3273	. . . . 5 ⊢ ((1st ‘𝑤)(1Walks‘𝐺)(2nd ‘𝑤) → ∃𝑓 𝑓(1Walks‘𝐺)(2nd ‘𝑤))
4		1wlkiswwlks 41073	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (∃𝑓 𝑓(1Walks‘𝐺)(2nd ‘𝑤) ↔ (2nd ‘𝑤) ∈ (WWalkS‘𝐺)))
5	3, 4	syl5ib 233	. . . 4 ⊢ (𝐺 ∈ USPGraph → ((1st ‘𝑤)(1Walks‘𝐺)(2nd ‘𝑤) → (2nd ‘𝑤) ∈ (WWalkS‘𝐺)))
6		1wlkcpr 40833	. . . . 5 ⊢ (𝑤 ∈ (1Walks‘𝐺) ↔ (1st ‘𝑤)(1Walks‘𝐺)(2nd ‘𝑤))
7	6	biimpi 205	. . . 4 ⊢ (𝑤 ∈ (1Walks‘𝐺) → (1st ‘𝑤)(1Walks‘𝐺)(2nd ‘𝑤))
8	5, 7	impel 484	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ (1Walks‘𝐺)) → (2nd ‘𝑤) ∈ (WWalkS‘𝐺))
9		1wlkpwwlkf1ouspgr.f	. . 3 ⊢ 𝐹 = (𝑤 ∈ (1Walks‘𝐺) ↦ (2nd ‘𝑤))
10	8, 9	fmptd 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))
12	9	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ (1Walks‘𝐺) → 𝐹 = (𝑤 ∈ (1Walks‘𝐺) ↦ (2nd ‘𝑤)))
13		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (2nd ‘𝑤) = (2nd ‘𝑥))
14	13	adantl 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ (1Walks‘𝐺) ∧ 𝑤 = 𝑥) → (2nd ‘𝑤) = (2nd ‘𝑥))
15		id 22	. . . . . . . . 9 ⊢ (𝑥 ∈ (1Walks‘𝐺) → 𝑥 ∈ (1Walks‘𝐺))
16		fvex 6113	. . . . . . . . . 10 ⊢ (2nd ‘𝑥) ∈ V
17	16	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ (1Walks‘𝐺) → (2nd ‘𝑥) ∈ V)
18	12, 14, 15, 17	fvmptd 6197	. . . . . . . 8 ⊢ (𝑥 ∈ (1Walks‘𝐺) → (𝐹‘𝑥) = (2nd ‘𝑥))
19	9	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ∈ (1Walks‘𝐺) → 𝐹 = (𝑤 ∈ (1Walks‘𝐺) ↦ (2nd ‘𝑤)))
20		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (2nd ‘𝑤) = (2nd ‘𝑦))
21	20	adantl 481	. . . . . . . . 9 ⊢ ((𝑦 ∈ (1Walks‘𝐺) ∧ 𝑤 = 𝑦) → (2nd ‘𝑤) = (2nd ‘𝑦))
22		id 22	. . . . . . . . 9 ⊢ (𝑦 ∈ (1Walks‘𝐺) → 𝑦 ∈ (1Walks‘𝐺))
23		fvex 6113	. . . . . . . . . 10 ⊢ (2nd ‘𝑦) ∈ V
24	23	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ∈ (1Walks‘𝐺) → (2nd ‘𝑦) ∈ V)
25	19, 21, 22, 24	fvmptd 6197	. . . . . . . 8 ⊢ (𝑦 ∈ (1Walks‘𝐺) → (𝐹‘𝑦) = (2nd ‘𝑦))
26	18, 25	eqeqan12d 2626	. . . . . . 7 ⊢ ((𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 ∈ (1Walks‘𝐺)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (2nd ‘𝑥) = (2nd ‘𝑦)))
27	26	adantl 481	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) ∧ (𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 ∈ (1Walks‘𝐺))) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (2nd ‘𝑥) = (2nd ‘𝑦)))
28		uspgr2wlkeqi 40856	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 ∈ (1Walks‘𝐺)) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) → 𝑥 = 𝑦)
29	28	ad4ant134 1288	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) ∧ (𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 ∈ (1Walks‘𝐺))) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) → 𝑥 = 𝑦)
30	29	ex 449	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) ∧ (𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 ∈ (1Walks‘𝐺))) → ((2nd ‘𝑥) = (2nd ‘𝑦) → 𝑥 = 𝑦))
31	27, 30	sylbid 229	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) ∧ (𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 ∈ (1Walks‘𝐺))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
32	31	ralrimivva 2954	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) → ∀𝑥 ∈ (1Walks‘𝐺)∀𝑦 ∈ (1Walks‘𝐺)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
33		dff13 6416	. . . 4 ⊢ (𝐹:(1Walks‘𝐺)–1-1→(WWalkS‘𝐺) ↔ (𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺) ∧ ∀𝑥 ∈ (1Walks‘𝐺)∀𝑦 ∈ (1Walks‘𝐺)((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
34	11, 32, 33	sylanbrc 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)𝑦 ↔ 𝑦 ∈ (WWalkS‘𝐺)))
36	35	adantr 480	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) → (∃𝑓 𝑓(1Walks‘𝐺)𝑦 ↔ 𝑦 ∈ (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
40	38, 39	op2nd 7068	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨𝑓, 𝑦⟩) = 𝑦
41	40	eqcomi 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⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝑓, 𝑦⟩))
45	44	eqeq2d 2620	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → (𝑦 = (2nd ‘𝑥) ↔ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)))
46	43, 45	anbi12d 743	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝑓, 𝑦⟩ → ((𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)) ↔ (⟨𝑓, 𝑦⟩ ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩))))
47	42, 46	spcev 3273	. . . . . . . . . . . 12 ⊢ ((⟨𝑓, 𝑦⟩ ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd ‘⟨𝑓, 𝑦⟩)) → ∃𝑥(𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
48	41, 47	mpan2 703	. . . . . . . . . . 11 ⊢ (⟨𝑓, 𝑦⟩ ∈ (1Walks‘𝐺) → ∃𝑥(𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
49	37, 48	sylbi 206	. . . . . . . . . 10 ⊢ (𝑓(1Walks‘𝐺)𝑦 → ∃𝑥(𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
50	49	exlimiv 1845	. . . . . . . . 9 ⊢ (∃𝑓 𝑓(1Walks‘𝐺)𝑦 → ∃𝑥(𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
51	36, 50	syl6bir 243	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) → (𝑦 ∈ (WWalkS‘𝐺) → ∃𝑥(𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥))))
52	51	imp 444	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) ∧ 𝑦 ∈ (WWalkS‘𝐺)) → ∃𝑥(𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
53		df-rex 2902	. . . . . . 7 ⊢ (∃𝑥 ∈ (1Walks‘𝐺)𝑦 = (2nd ‘𝑥) ↔ ∃𝑥(𝑥 ∈ (1Walks‘𝐺) ∧ 𝑦 = (2nd ‘𝑥)))
54	52, 53	sylibr 223	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) ∧ 𝑦 ∈ (WWalkS‘𝐺)) → ∃𝑥 ∈ (1Walks‘𝐺)𝑦 = (2nd ‘𝑥))
55	18	eqeq2d 2620	. . . . . . 7 ⊢ (𝑥 ∈ (1Walks‘𝐺) → (𝑦 = (𝐹‘𝑥) ↔ 𝑦 = (2nd ‘𝑥)))
56	55	rexbiia 3022	. . . . . 6 ⊢ (∃𝑥 ∈ (1Walks‘𝐺)𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ (1Walks‘𝐺)𝑦 = (2nd ‘𝑥))
57	54, 56	sylibr 223	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) ∧ 𝑦 ∈ (WWalkS‘𝐺)) → ∃𝑥 ∈ (1Walks‘𝐺)𝑦 = (𝐹‘𝑥))
58	57	ralrimiva 2949	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) → ∀𝑦 ∈ (WWalkS‘𝐺)∃𝑥 ∈ (1Walks‘𝐺)𝑦 = (𝐹‘𝑥))
59		dffo3 6282	. . . 4 ⊢ (𝐹:(1Walks‘𝐺)–onto→(WWalkS‘𝐺) ↔ (𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺) ∧ ∀𝑦 ∈ (WWalkS‘𝐺)∃𝑥 ∈ (1Walks‘𝐺)𝑦 = (𝐹‘𝑥)))
60	11, 58, 59	sylanbrc 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)))
62	34, 60, 61	sylanbrc 695	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹:(1Walks‘𝐺)⟶(WWalkS‘𝐺)) → 𝐹:(1Walks‘𝐺)–1-1-onto→(WWalkS‘𝐺))
63	10, 62	mpdan 699	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:(1Walks‘𝐺)–1-1-onto→(WWalkS‘𝐺))

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  1^st c1st 7057  2^nd 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