Theorem wspthnp 30140

Description: Properties of a set being a simple path of a fixed length as word. (Contributed by AV, 18-May-2021.)

Assertion

Ref	Expression
wspthnp	⊢ (𝑊 ∈ (𝑁 WSPathsN 𝐺) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))

Distinct variable groups:   𝑓,𝐺   𝑓,𝑊

Allowed substitution hint:   𝑁(𝑓)

Proof of Theorem wspthnp

Dummy variables 𝑔 𝑛 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wspthsn 30123	. . 3 ⊢ WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤})
2	1	elmpocl 7652	. 2 ⊢ (𝑊 ∈ (𝑁 WSPathsN 𝐺) → (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V))
3		simpl 487	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WSPathsN 𝐺)) → (𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V))
4		iswspthn 30139	. . . 4 ⊢ (𝑊 ∈ (𝑁 WSPathsN 𝐺) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))
5	4	bilani 509	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WSPathsN 𝐺)) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))
6		3anass 1109	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊) ↔ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊)))
7	3, 5, 6	sylanbrc 594	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WSPathsN 𝐺)) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))
8	2, 7	mpancom 700	1 ⊢ (𝑊 ∈ (𝑁 WSPathsN 𝐺) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101  ∃wex 1806   ∈ wcel 2149  {crab 3422  Vcvv 3461   class class class wbr 5111  ‘cfv 6537  (class class class)co 7411  ℕ₀cn0 12504  SPathscspths 30001   WWalksN cwwlksn 30116   WSPathsN cwwspthsn 30118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-sbc 3752  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-opab 5176  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-wwlksn 30121  df-wspthsn 30123

This theorem is referenced by:  wspthsswwlkn  30208