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Metamath Proof Explorer

Theorem vtxdgfival

Description: The degree of a vertex for graphs of finite size. (Contributed by Mario Carneiro, 12-Mar-2015) (Revised by Alexander van der Vekens, 21-Jan-2018) (Revised by AV, 8-Dec-2020) (Revised by AV, 22-Mar-2021)

Ref Expression
Hypotheses vtxdgval.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
vtxdgval.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
vtxdgval.a ⊢ 𝐴 = dom 𝐼
Assertion vtxdgfival ( ( 𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉 ) → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑈 ) = ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) + ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ) )

Proof

Step Hyp Ref Expression
1 vtxdgval.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2 vtxdgval.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
3 vtxdgval.a ⊢ 𝐴 = dom 𝐼
4 1 2 3 vtxdgval ⊢ ( 𝑈 ∈ 𝑉 → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑈 ) = ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ) )
5 4 adantl ⊢ ( ( 𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉 ) → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑈 ) = ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ) )
6 rabfi ⊢ ( 𝐴 ∈ Fin → { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ∈ Fin )
7 hashcl ⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ∈ Fin → ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) ∈ ℕ0 )
8 6 7 syl ⊢ ( 𝐴 ∈ Fin → ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) ∈ ℕ0 )
9 8 nn0red ⊢ ( 𝐴 ∈ Fin → ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) ∈ ℝ )
10 rabfi ⊢ ( 𝐴 ∈ Fin → { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ∈ Fin )
11 hashcl ⊢ ( { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ∈ Fin → ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ∈ ℕ0 )
12 10 11 syl ⊢ ( 𝐴 ∈ Fin → ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ∈ ℕ0 )
13 12 nn0red ⊢ ( 𝐴 ∈ Fin → ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ∈ ℝ )
14 9 13 jca ⊢ ( 𝐴 ∈ Fin → ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) ∈ ℝ ∧ ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ∈ ℝ ) )
15 14 adantr ⊢ ( ( 𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉 ) → ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) ∈ ℝ ∧ ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ∈ ℝ ) )
16 rexadd ⊢ ( ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) ∈ ℝ ∧ ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ∈ ℝ ) → ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ) = ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) + ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ) )
17 15 16 syl ⊢ ( ( 𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉 ) → ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) +𝑒 ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ) = ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) + ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ) )
18 5 17 eqtrd ⊢ ( ( 𝐴 ∈ Fin ∧ 𝑈 ∈ 𝑉 ) → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑈 ) = ( ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ 𝑈 ∈ ( 𝐼 ‘ 𝑥 ) } ) + ( ♯ ‘ { 𝑥 ∈ 𝐴 ∣ ( 𝐼 ‘ 𝑥 ) = { 𝑈 } } ) ) )