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Metamath Proof Explorer

Theorem usgr2wlkspthlem1

Description: Lemma 1 for usgr2wlkspth . (Contributed by Alexander van der Vekens, 2-Mar-2018) (Revised by AV, 26-Jan-2021)

Ref Expression
Assertion usgr2wlkspthlem1 ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) → Fun ◡ 𝐹 )

Proof

Step Hyp Ref Expression
1 simp1 ⊢ ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → 𝐺 ∈ USGraph )
2 1 anim2i ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝐺 ∈ USGraph ) )
3 2 ancomd ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝐺 ∈ USGraph ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) )
4 3simpc ⊢ ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → ( ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) )
5 4 adantl ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) )
6 usgr2wlkneq ⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) ∧ ( ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) )
7 3 5 6 syl2anc ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) )
8 fvexd ⊢ ( ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) → ( 𝐹 ‘ 0 ) ∈ V )
9 fvexd ⊢ ( ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) → ( 𝐹 ‘ 1 ) ∈ V )
10 simpr ⊢ ( ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) → ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) )
11 8 9 10 3jca ⊢ ( ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) → ( ( 𝐹 ‘ 0 ) ∈ V ∧ ( 𝐹 ‘ 1 ) ∈ V ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) )
12 funcnvs2 ⊢ ( ( ( 𝐹 ‘ 0 ) ∈ V ∧ ( 𝐹 ‘ 1 ) ∈ V ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) → Fun ◡ ⟨“ ( 𝐹 ‘ 0 ) ( 𝐹 ‘ 1 ) ”⟩ )
13 7 11 12 3syl ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) → Fun ◡ ⟨“ ( 𝐹 ‘ 0 ) ( 𝐹 ‘ 1 ) ”⟩ )
14 eqid ⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
15 14 wlkf ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) )
16 simp2 ⊢ ( ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → ( ♯ ‘ 𝐹 ) = 2 )
17 wrdlen2s2 ⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ ( ♯ ‘ 𝐹 ) = 2 ) → 𝐹 = ⟨“ ( 𝐹 ‘ 0 ) ( 𝐹 ‘ 1 ) ”⟩ )
18 15 16 17 syl2an ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) → 𝐹 = ⟨“ ( 𝐹 ‘ 0 ) ( 𝐹 ‘ 1 ) ”⟩ )
19 18 cnveqd ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) → ◡ 𝐹 = ◡ ⟨“ ( 𝐹 ‘ 0 ) ( 𝐹 ‘ 1 ) ”⟩ )
20 19 funeqd ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) → ( Fun ◡ 𝐹 ↔ Fun ◡ ⟨“ ( 𝐹 ‘ 0 ) ( 𝐹 ‘ 1 ) ”⟩ ) )
21 13 20 mpbird ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝐺 ∈ USGraph ∧ ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) → Fun ◡ 𝐹 )