Theorem crcts 29874

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 262	. 2 ⊢ (𝑔 = 𝐺 → ((𝑝‘0) = (𝑝‘(♯‘𝑓)) ↔ (𝑝‘0) = (𝑝‘(♯‘𝑓))))
2		df-crcts 29872	. 2 ⊢ Circuits = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
3	1, 2	fvmptopab 7416	1 ⊢ (Circuits‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}

Syntax hints:   ∧ wa 395   = wceq 1542   class class class wbr 5086  {copab 5148  ‘cfv 6493  0cc0 11032  ♯chash 14286  Trailsctrls 29775  Circuitsccrcts 29870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-crcts 29872

This theorem is referenced by:  iscrct  29876