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Metamath Proof Explorer

Theorem wlkiswwlks2lem4

Description: Lemma 4 for wlkiswwlks2 . (Contributed by Alexander van der Vekens, 20-Jul-2018) (Revised by AV, 10-Apr-2021)

Ref Expression
Hypotheses wlkiswwlks2lem.f ⊢ 𝐹 = ( 𝑥 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ↦ ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ ( 𝑥 + 1 ) ) } ) )
wlkiswwlks2lem.e ⊢ 𝐸 = ( iEdg ‘ 𝐺 )
Assertion wlkiswwlks2lem4 ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ ( ♯ ‘ 𝑃 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ran 𝐸 → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐸 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) )

Proof

Step Hyp Ref Expression
1 wlkiswwlks2lem.f ⊢ 𝐹 = ( 𝑥 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ↦ ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ ( 𝑥 + 1 ) ) } ) )
2 wlkiswwlks2lem.e ⊢ 𝐸 = ( iEdg ‘ 𝐺 )
3 1 wlkiswwlks2lem1 ⊢ ( ( 𝑃 ∈ Word 𝑉 ∧ 1 ≤ ( ♯ ‘ 𝑃 ) ) → ( ♯ ‘ 𝐹 ) = ( ( ♯ ‘ 𝑃 ) − 1 ) )
4 3 3adant1 ⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ ( ♯ ‘ 𝑃 ) ) → ( ♯ ‘ 𝐹 ) = ( ( ♯ ‘ 𝑃 ) − 1 ) )
5 lencl ⊢ ( 𝑃 ∈ Word 𝑉 → ( ♯ ‘ 𝑃 ) ∈ ℕ0 )
6 5 3ad2ant2 ⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ ( ♯ ‘ 𝑃 ) ) → ( ♯ ‘ 𝑃 ) ∈ ℕ0 )
7 1 wlkiswwlks2lem2 ⊢ ( ( ( ♯ ‘ 𝑃 ) ∈ ℕ0 ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ) → ( 𝐹 ‘ 𝑖 ) = ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) )
8 6 7 sylan ⊢ ( ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ) → ( 𝐹 ‘ 𝑖 ) = ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) )
9 8 adantr ⊢ ( ( ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ) ∧ { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ran 𝐸 ) → ( 𝐹 ‘ 𝑖 ) = ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) )
10 9 fveq2d ⊢ ( ( ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ) ∧ { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ran 𝐸 ) → ( 𝐸 ‘ ( 𝐹 ‘ 𝑖 ) ) = ( 𝐸 ‘ ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) ) )
11 2 uspgrf1oedg ⊢ ( 𝐺 ∈ USPGraph → 𝐸 : dom 𝐸 –1-1-onto→ ( Edg ‘ 𝐺 ) )
12 2 rneqi ⊢ ran 𝐸 = ran ( iEdg ‘ 𝐺 )
13 edgval ⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 )
14 12 13 eqtr4i ⊢ ran 𝐸 = ( Edg ‘ 𝐺 )
15 f1oeq3 ⊢ ( ran 𝐸 = ( Edg ‘ 𝐺 ) → ( 𝐸 : dom 𝐸 –1-1-onto→ ran 𝐸 ↔ 𝐸 : dom 𝐸 –1-1-onto→ ( Edg ‘ 𝐺 ) ) )
16 14 15 ax-mp ⊢ ( 𝐸 : dom 𝐸 –1-1-onto→ ran 𝐸 ↔ 𝐸 : dom 𝐸 –1-1-onto→ ( Edg ‘ 𝐺 ) )
17 11 16 sylibr ⊢ ( 𝐺 ∈ USPGraph → 𝐸 : dom 𝐸 –1-1-onto→ ran 𝐸 )
18 17 3ad2ant1 ⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ ( ♯ ‘ 𝑃 ) ) → 𝐸 : dom 𝐸 –1-1-onto→ ran 𝐸 )
19 18 adantr ⊢ ( ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ) → 𝐸 : dom 𝐸 –1-1-onto→ ran 𝐸 )
20 f1ocnvfv2 ⊢ ( ( 𝐸 : dom 𝐸 –1-1-onto→ ran 𝐸 ∧ { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ran 𝐸 ) → ( 𝐸 ‘ ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } )
21 19 20 sylan ⊢ ( ( ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ) ∧ { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ran 𝐸 ) → ( 𝐸 ‘ ( ◡ 𝐸 ‘ { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } )
22 10 21 eqtrd ⊢ ( ( ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ) ∧ { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ran 𝐸 ) → ( 𝐸 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } )
23 22 ex ⊢ ( ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ) → ( { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ran 𝐸 → ( 𝐸 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) )
24 23 ralimdva ⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ ( ♯ ‘ 𝑃 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ran 𝐸 → ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ( 𝐸 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) )
25 oveq2 ⊢ ( ( ♯ ‘ 𝐹 ) = ( ( ♯ ‘ 𝑃 ) − 1 ) → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) )
26 25 raleqdv ⊢ ( ( ♯ ‘ 𝐹 ) = ( ( ♯ ‘ 𝑃 ) − 1 ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐸 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ↔ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ( 𝐸 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) )
27 26 imbi2d ⊢ ( ( ♯ ‘ 𝐹 ) = ( ( ♯ ‘ 𝑃 ) − 1 ) → ( ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ran 𝐸 → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐸 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) ↔ ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ran 𝐸 → ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) ( 𝐸 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) ) )
28 24 27 imbitrrid ⊢ ( ( ♯ ‘ 𝐹 ) = ( ( ♯ ‘ 𝑃 ) − 1 ) → ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ ( ♯ ‘ 𝑃 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ran 𝐸 → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐸 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) ) )
29 4 28 mpcom ⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ ( ♯ ‘ 𝑃 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ran 𝐸 → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐸 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) )