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Metamath Proof Explorer

Theorem usgrsscusgr

Description: A simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018) (Revised by AV, 13-Nov-2020)

Ref Expression
Hypotheses fusgrmaxsize.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
fusgrmaxsize.e ⊢ 𝐸 = ( Edg ‘ 𝐺 )
usgrsscusgra.h ⊢ 𝑉 = ( Vtx ‘ 𝐻 )
usgrsscusgra.f ⊢ 𝐹 = ( Edg ‘ 𝐻 )
Assertion usgrsscusgr ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph ) → ∀ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 𝑒 = 𝑓 )

Proof

Step Hyp Ref Expression
1 fusgrmaxsize.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2 fusgrmaxsize.e ⊢ 𝐸 = ( Edg ‘ 𝐺 )
3 usgrsscusgra.h ⊢ 𝑉 = ( Vtx ‘ 𝐻 )
4 usgrsscusgra.f ⊢ 𝐹 = ( Edg ‘ 𝐻 )
5 1 2 3 4 usgredgsscusgredg ⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph ) → 𝐸 ⊆ 𝐹 )
6 dfss5 ⊢ ( 𝐸 ⊆ 𝐹 ↔ ∀ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 𝑒 = 𝑓 )
7 5 6 sylib ⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph ) → ∀ 𝑒 ∈ 𝐸 ∃ 𝑓 ∈ 𝐹 𝑒 = 𝑓 )