Theorem crctiswlk 29882

Description: A circuit is a walk. (Contributed by AV, 6-Apr-2021.)

Assertion

Ref	Expression
crctiswlk	⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)

Proof of Theorem crctiswlk

Step	Hyp	Ref	Expression
1		crctistrl 29881	. 2 ⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)
2		trliswlk 29782	. 2 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
3	1, 2	syl 17	1 ⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)

Syntax hints:   → wi 4   class class class wbr 5072  ‘cfv 6485  Walkscwlks 29683  Trailsctrls 29775  Circuitsccrcts 29870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fv 6493  df-wlks 29686  df-trls 29777  df-crcts 29872

This theorem is referenced by:  crctcshlem1  29903  crctcshlem2  29904  crctcshlem4  29906  crctcshwlkn0  29907  eucrctshift  30331  eucrct2eupth  30333