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Theorem grtri 47845
Description: The triangles in a graph. (Contributed by AV, 20-Jul-2025.)
Hypotheses
Ref Expression
grtri.v 𝑉 = (Vtx‘𝐺)
grtri.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
grtri (𝐺𝑊 → (GrTriangles‘𝐺) = {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))})
Distinct variable groups:   𝑓,𝐸,𝑡   𝑓,𝐺,𝑡   𝑓,𝑉,𝑡
Allowed substitution hints:   𝑊(𝑡,𝑓)

Proof of Theorem grtri
Dummy variables 𝑒 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-grtri 47843 . . 3 GrTriangles = (𝑔 ∈ V ↦ (Vtx‘𝑔) / 𝑣(Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))})
21a1i 11 . 2 (𝐺𝑊 → GrTriangles = (𝑔 ∈ V ↦ (Vtx‘𝑔) / 𝑣(Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))}))
3 fveq2 6907 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4 grtri.v . . . . . 6 𝑉 = (Vtx‘𝐺)
53, 4eqtr4di 2793 . . . . 5 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
6 fveq2 6907 . . . . . . 7 (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
7 grtri.e . . . . . . 7 𝐸 = (Edg‘𝐺)
86, 7eqtr4di 2793 . . . . . 6 (𝑔 = 𝐺 → (Edg‘𝑔) = 𝐸)
98csbeq1d 3912 . . . . 5 (𝑔 = 𝐺(Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))} = 𝐸 / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))})
105, 9csbeq12dv 3917 . . . 4 (𝑔 = 𝐺(Vtx‘𝑔) / 𝑣(Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))} = 𝑉 / 𝑣𝐸 / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))})
1110adantl 481 . . 3 ((𝐺𝑊𝑔 = 𝐺) → (Vtx‘𝑔) / 𝑣(Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))} = 𝑉 / 𝑣𝐸 / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))})
124fvexi 6921 . . . 4 𝑉 ∈ V
137fvexi 6921 . . . 4 𝐸 ∈ V
14 pweq 4619 . . . . . 6 (𝑣 = 𝑉 → 𝒫 𝑣 = 𝒫 𝑉)
1514adantr 480 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝒫 𝑣 = 𝒫 𝑉)
16 eleq2 2828 . . . . . . . . 9 (𝑒 = 𝐸 → ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ↔ {(𝑓‘0), (𝑓‘1)} ∈ 𝐸))
17 eleq2 2828 . . . . . . . . 9 (𝑒 = 𝐸 → ({(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ↔ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸))
18 eleq2 2828 . . . . . . . . 9 (𝑒 = 𝐸 → ({(𝑓‘1), (𝑓‘2)} ∈ 𝑒 ↔ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))
1916, 17, 183anbi123d 1435 . . . . . . . 8 (𝑒 = 𝐸 → (({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒) ↔ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)))
2019anbi2d 630 . . . . . . 7 (𝑒 = 𝐸 → ((𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒)) ↔ (𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))))
2120exbidv 1919 . . . . . 6 (𝑒 = 𝐸 → (∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒)) ↔ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))))
2221adantl 481 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒)) ↔ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))))
2315, 22rabeqbidv 3452 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → {𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))} = {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))})
2412, 13, 23csbie2 3950 . . 3 𝑉 / 𝑣𝐸 / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))} = {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))}
2511, 24eqtrdi 2791 . 2 ((𝐺𝑊𝑔 = 𝐺) → (Vtx‘𝑔) / 𝑣(Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))} = {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))})
26 elex 3499 . 2 (𝐺𝑊𝐺 ∈ V)
274pweqi 4621 . . . . 5 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
28 fvex 6920 . . . . . 6 (Vtx‘𝐺) ∈ V
2928pwex 5386 . . . . 5 𝒫 (Vtx‘𝐺) ∈ V
3027, 29eqeltri 2835 . . . 4 𝒫 𝑉 ∈ V
3130rabex 5345 . . 3 {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))} ∈ V
3231a1i 11 . 2 (𝐺𝑊 → {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))} ∈ V)
332, 25, 26, 32fvmptd 7023 1 (𝐺𝑊 → (GrTriangles‘𝐺) = {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {crab 3433  Vcvv 3478  csb 3908  𝒫 cpw 4605  {cpr 4633  cmpt 5231  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154  2c2 12319  3c3 12320  ..^cfzo 13691  Vtxcvtx 29028  Edgcedg 29079  GrTrianglescgrtri 47842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-grtri 47843
This theorem is referenced by:  grtriprop  47846  isgrtri  47848
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