Theorem wspthnon 26743

Description: An element of the set of simple paths of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 12-May-2021.)

Hypothesis

Ref	Expression
wwlknon.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
wspthnon	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)))

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝑓,𝐺   𝑓,𝑁   𝑓,𝑊

Allowed substitution hint:   𝑉(𝑓)

Proof of Theorem wspthnon

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlknon.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	iswspthsnon 26741	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})
3	2	eleq2d 2687	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ↔ 𝑊 ∈ {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤}))
4		breq2 4657	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤 ↔ 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊))
5	4	exbidv 1850	. . 3 ⊢ (𝑤 = 𝑊 → (∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤 ↔ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊))
6	5	elrab 3363	. 2 ⊢ (𝑊 ∈ {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} ↔ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊))
7	3, 6	syl6bb 276	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {crab 2916   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  Vtxcvtx 25874  SPathsOncspthson 26611   WWalksNOn cwwlksnon 26719   WSPathsNOn cwwspthsnon 26721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-wwlksnon 26724  df-wspthsnon 26726

This theorem is referenced by:  wspthnonp  26744  wspthsnwspthsnon  26811  elwspths2on  26853  elwspths2spth  26862