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Theorem grtriprop 47908
Description: The properties of a triangle. (Contributed by AV, 25-Jul-2025.)
Hypotheses
Ref Expression
grtri.v 𝑉 = (Vtx‘𝐺)
grtri.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
grtriprop (𝑇 ∈ (GrTriangles‘𝐺) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
Distinct variable groups:   𝑥,𝐸,𝑦,𝑧   𝑥,𝑇,𝑦,𝑧   𝑥,𝑉,𝑦,𝑧
Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧)

Proof of Theorem grtriprop
Dummy variables 𝑓 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6944 . . . . . 6 (𝑇 ∈ (GrTriangles‘𝐺) → 𝐺 ∈ V)
2 grtri.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
3 grtri.e . . . . . . 7 𝐸 = (Edg‘𝐺)
42, 3grtri 47907 . . . . . 6 (𝐺 ∈ V → (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))})
51, 4syl 17 . . . . 5 (𝑇 ∈ (GrTriangles‘𝐺) → (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))})
65eleq2d 2827 . . . 4 (𝑇 ∈ (GrTriangles‘𝐺) → (𝑇 ∈ (GrTriangles‘𝐺) ↔ 𝑇 ∈ {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))}))
7 f1oeq3 6838 . . . . . . 7 (𝑡 = 𝑇 → (𝑓:(0..^3)–1-1-onto𝑡𝑓:(0..^3)–1-1-onto𝑇))
87anbi1d 631 . . . . . 6 (𝑡 = 𝑇 → ((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))))
98exbidv 1921 . . . . 5 (𝑡 = 𝑇 → (∃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))))
109elrab 3692 . . . 4 (𝑇 ∈ {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))))
116, 10bitrdi 287 . . 3 (𝑇 ∈ (GrTriangles‘𝐺) → (𝑇 ∈ (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)))))
12 ovexd 7466 . . . . . . . 8 ((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) → (0..^3) ∈ V)
13 simpr 484 . . . . . . . 8 ((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) → 𝑓:(0..^3)–1-1-onto𝑇)
1412, 13hasheqf1od 14392 . . . . . . 7 ((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) → (♯‘(0..^3)) = (♯‘𝑇))
15 eqcom 2744 . . . . . . . . 9 ((♯‘(0..^3)) = (♯‘𝑇) ↔ (♯‘𝑇) = (♯‘(0..^3)))
16 3nn0 12544 . . . . . . . . . . 11 3 ∈ ℕ0
17 hashfzo0 14469 . . . . . . . . . . 11 (3 ∈ ℕ0 → (♯‘(0..^3)) = 3)
1816, 17mp1i 13 . . . . . . . . . 10 ((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) → (♯‘(0..^3)) = 3)
1918eqeq2d 2748 . . . . . . . . 9 ((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) → ((♯‘𝑇) = (♯‘(0..^3)) ↔ (♯‘𝑇) = 3))
2015, 19bitrid 283 . . . . . . . 8 ((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) → ((♯‘(0..^3)) = (♯‘𝑇) ↔ (♯‘𝑇) = 3))
21 hash3tpb 14534 . . . . . . . . . . . 12 (𝑇 ∈ 𝒫 𝑉 → ((♯‘𝑇) = 3 ↔ ∃𝑥𝑇𝑦𝑇𝑧𝑇 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
2221adantr 480 . . . . . . . . . . 11 ((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) → ((♯‘𝑇) = 3 ↔ ∃𝑥𝑇𝑦𝑇𝑧𝑇 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
2322biimpa 476 . . . . . . . . . 10 (((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) ∧ (♯‘𝑇) = 3) → ∃𝑥𝑇𝑦𝑇𝑧𝑇 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}))
24 elpwi 4607 . . . . . . . . . . . . . 14 (𝑇 ∈ 𝒫 𝑉𝑇𝑉)
25 ss2rexv 4055 . . . . . . . . . . . . . . 15 (𝑇𝑉 → (∃𝑥𝑇𝑦𝑇𝑧𝑇 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑥𝑉𝑦𝑉𝑧𝑇 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
26 ssrexv 4053 . . . . . . . . . . . . . . . . 17 (𝑇𝑉 → (∃𝑧𝑇 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑧𝑉 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
2726reximdv 3170 . . . . . . . . . . . . . . . 16 (𝑇𝑉 → (∃𝑦𝑉𝑧𝑇 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑦𝑉𝑧𝑉 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
2827reximdv 3170 . . . . . . . . . . . . . . 15 (𝑇𝑉 → (∃𝑥𝑉𝑦𝑉𝑧𝑇 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
2925, 28syld 47 . . . . . . . . . . . . . 14 (𝑇𝑉 → (∃𝑥𝑇𝑦𝑇𝑧𝑇 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
3024, 29syl 17 . . . . . . . . . . . . 13 (𝑇 ∈ 𝒫 𝑉 → (∃𝑥𝑇𝑦𝑇𝑧𝑇 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
3130adantr 480 . . . . . . . . . . . 12 ((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) → (∃𝑥𝑇𝑦𝑇𝑧𝑇 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
3231adantr 480 . . . . . . . . . . 11 (((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) ∧ (♯‘𝑇) = 3) → (∃𝑥𝑇𝑦𝑇𝑧𝑇 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})))
33 simprr 773 . . . . . . . . . . . . . . . . 17 (((((((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑧𝑉) ∧ ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})) → 𝑇 = {𝑥, 𝑦, 𝑧})
34 simp-5r 786 . . . . . . . . . . . . . . . . 17 (((((((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑧𝑉) ∧ ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})) → (♯‘𝑇) = 3)
35 f1oeq3 6838 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑇 = {𝑥, 𝑦, 𝑧} → (𝑓:(0..^3)–1-1-onto𝑇𝑓:(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧}))
36 grtriproplem 47906 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓:(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧} ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))
37362a1d 26 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓:(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧} ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) → ((𝑥𝑉𝑦𝑉) → (𝑧𝑉 → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
3837ex 412 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓:(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧} → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ((𝑥𝑉𝑦𝑉) → (𝑧𝑉 → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))
3938a1d 25 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓:(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧} → ((♯‘𝑇) = 3 → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ((𝑥𝑉𝑦𝑉) → (𝑧𝑉 → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))))
4035, 39biimtrdi 253 . . . . . . . . . . . . . . . . . . . . . 22 (𝑇 = {𝑥, 𝑦, 𝑧} → (𝑓:(0..^3)–1-1-onto𝑇 → ((♯‘𝑇) = 3 → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ((𝑥𝑉𝑦𝑉) → (𝑧𝑉 → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))))
4140adantld 490 . . . . . . . . . . . . . . . . . . . . 21 (𝑇 = {𝑥, 𝑦, 𝑧} → ((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) → ((♯‘𝑇) = 3 → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ((𝑥𝑉𝑦𝑉) → (𝑧𝑉 → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))))
4241imp4c 423 . . . . . . . . . . . . . . . . . . . 20 (𝑇 = {𝑥, 𝑦, 𝑧} → ((((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) → ((𝑥𝑉𝑦𝑉) → (𝑧𝑉 → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))
4342imp4c 423 . . . . . . . . . . . . . . . . . . 19 (𝑇 = {𝑥, 𝑦, 𝑧} → ((((((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑧𝑉) → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
4443adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ((((((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑧𝑉) → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
4544impcom 407 . . . . . . . . . . . . . . . . 17 (((((((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑧𝑉) ∧ ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})) → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))
4633, 34, 453jca 1129 . . . . . . . . . . . . . . . 16 (((((((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑧𝑉) ∧ ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧})) → (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
4746ex 412 . . . . . . . . . . . . . . 15 ((((((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ∧ (𝑥𝑉𝑦𝑉)) ∧ 𝑧𝑉) → (((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
4847reximdva 3168 . . . . . . . . . . . . . 14 (((((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑧𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
4948reximdvva 3207 . . . . . . . . . . . . 13 ((((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) ∧ (♯‘𝑇) = 3) ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
5049ex 412 . . . . . . . . . . . 12 (((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) ∧ (♯‘𝑇) = 3) → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))
5150com23 86 . . . . . . . . . . 11 (((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) ∧ (♯‘𝑇) = 3) → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))
5232, 51syld 47 . . . . . . . . . 10 (((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) ∧ (♯‘𝑇) = 3) → (∃𝑥𝑇𝑦𝑇𝑧𝑇 ((𝑥𝑦𝑥𝑧𝑦𝑧) ∧ 𝑇 = {𝑥, 𝑦, 𝑧}) → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))
5323, 52mpd 15 . . . . . . . . 9 (((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) ∧ (♯‘𝑇) = 3) → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
5453ex 412 . . . . . . . 8 ((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) → ((♯‘𝑇) = 3 → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))
5520, 54sylbid 240 . . . . . . 7 ((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) → ((♯‘(0..^3)) = (♯‘𝑇) → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))))
5614, 55mpd 15 . . . . . 6 ((𝑇 ∈ 𝒫 𝑉𝑓:(0..^3)–1-1-onto𝑇) → (({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
5756expimpd 453 . . . . 5 (𝑇 ∈ 𝒫 𝑉 → ((𝑓:(0..^3)–1-1-onto𝑇 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
5857exlimdv 1933 . . . 4 (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)) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
5958imp 406 . . 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))) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
6011, 59biimtrdi 253 . 2 (𝑇 ∈ (GrTriangles‘𝐺) → (𝑇 ∈ (GrTriangles‘𝐺) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))))
6160pm2.43i 52 1 (𝑇 ∈ (GrTriangles‘𝐺) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  wrex 3070  {crab 3436  Vcvv 3480  wss 3951  𝒫 cpw 4600  {cpr 4628  {ctp 4630  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156  2c2 12321  3c3 12322  0cn0 12526  ..^cfzo 13694  chash 14369  Vtxcvtx 29013  Edgcedg 29064  GrTrianglescgrtri 47904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-3o 8508  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-grtri 47905
This theorem is referenced by:  grtrif1o  47909  isgrtri  47910  grtrissvtx  47911  grimgrtri  47916  grlimgrtri  47963
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