Theorem crcts 29932

Description: The set of circuits (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)

Assertion

Ref	Expression
crcts	⊢ (Circuits‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}

Distinct variable group:   𝑓,𝐺,𝑝

Proof of Theorem crcts

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		biidd 264	. 2 ⊢ (𝑔 = 𝐺 → ((𝑝‘0) = (𝑝‘(♯‘𝑓)) ↔ (𝑝‘0) = (𝑝‘(♯‘𝑓))))
2		df-crcts 29930	. 2 ⊢ Circuits = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
3	1, 2	fvmptopab 7445	1 ⊢ (Circuits‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}

Syntax hints:   ∧ wa 399   = wceq 1559   class class class wbr 5099  {copab 5161  ‘cfv 6515  0cc0 11068  ♯chash 14338  Trailsctrls 29833  Circuitsccrcts 29928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6471  df-fun 6517  df-fv 6523  df-crcts 29930

This theorem is referenced by:  iscrct  29934