Theorem pthsfval 30310

Description: The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.)

Assertion

Ref	Expression
pthsfval	⊢ (Paths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)}

Distinct variable group:   𝑓,𝐺,𝑝

Proof of Theorem pthsfval

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		biidd 265	. . 3 ⊢ (𝑔 = 𝐺 → ((Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅) ↔ (Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)))
2		df-pths 30305	. . . 4 ⊢ Paths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)})
3		3anass 1111	. . . . . 6 ⊢ ((𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅) ↔ (𝑓(Trails‘𝑔)𝑝 ∧ (Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)))
4	3	opabbii 5171	. . . . 5 ⊢ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅))}
5	4	mpteq2i 5200	. . . 4 ⊢ (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)}) = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅))})
6	2, 5	eqtri 2783	. . 3 ⊢ Paths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅))})
7	1, 6	fvmptopab 7476	. 2 ⊢ (Paths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ (Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅))}
8		3anass 1111	. . 3 ⊢ ((𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅) ↔ (𝑓(Trails‘𝐺)𝑝 ∧ (Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)))
9	8	opabbii 5171	. 2 ⊢ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ (Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅))}
10	7, 9	eqtr4i 2786	1 ⊢ (Paths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)}

This proof depends on syntax axioms:   ∧ wa 401   ∧ w3a 1103   = wceq 1569  Vcvv 3450   ∩ cin 4089  ∅c0 4278  {cpr 4586   class class class wbr 5102  {copab 5166   ↦ cmpt 5185  ^◡ccnv 5649   ↾ cres 5652   “ cima 5653  Fun wfun 6533  ‘cfv 6539  (class class class)co 7421  0cc0 11204  1c1 11205  ..^cfzo 13792  ♯chash 14478  Trailsctrls 30279  Pathscpths 30301

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547  df-pths 30305

This theorem is used by:  ispth  30312