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Metamath Proof Explorer

Theorem 2wlkdlem6

Description: Lemma 6 for 2wlkd . (Contributed by AV, 23-Jan-2021)

Ref Expression
Hypotheses 2wlkd.p ⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 ”⟩
2wlkd.f ⊢ 𝐹 = ⟨“ 𝐽 𝐾 ”⟩
2wlkd.s ⊢ ( 𝜑 → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
2wlkd.n ⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) )
2wlkd.e ⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
Assertion 2wlkdlem6 ( 𝜑 → ( 𝐵 ∈ ( 𝐼 ‘ 𝐽 ) ∧ 𝐵 ∈ ( 𝐼 ‘ 𝐾 ) ) )

Proof

Step Hyp Ref Expression
1 2wlkd.p ⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 ”⟩
2 2wlkd.f ⊢ 𝐹 = ⟨“ 𝐽 𝐾 ”⟩
3 2wlkd.s ⊢ ( 𝜑 → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
4 2wlkd.n ⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) )
5 2wlkd.e ⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
6 prcom ⊢ { 𝐴 , 𝐵 } = { 𝐵 , 𝐴 }
7 6 sseq1i ⊢ ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ↔ { 𝐵 , 𝐴 } ⊆ ( 𝐼 ‘ 𝐽 ) )
8 7 biimpi ⊢ ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) → { 𝐵 , 𝐴 } ⊆ ( 𝐼 ‘ 𝐽 ) )
9 8 adantl ⊢ ( ( 𝜑 ∧ { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ) → { 𝐵 , 𝐴 } ⊆ ( 𝐼 ‘ 𝐽 ) )
10 3 simp2d ⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
11 3 simp1d ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
12 11 adantr ⊢ ( ( 𝜑 ∧ { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ) → 𝐴 ∈ 𝑉 )
13 prssg ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ) → ( ( 𝐵 ∈ ( 𝐼 ‘ 𝐽 ) ∧ 𝐴 ∈ ( 𝐼 ‘ 𝐽 ) ) ↔ { 𝐵 , 𝐴 } ⊆ ( 𝐼 ‘ 𝐽 ) ) )
14 10 12 13 syl2an2r ⊢ ( ( 𝜑 ∧ { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ) → ( ( 𝐵 ∈ ( 𝐼 ‘ 𝐽 ) ∧ 𝐴 ∈ ( 𝐼 ‘ 𝐽 ) ) ↔ { 𝐵 , 𝐴 } ⊆ ( 𝐼 ‘ 𝐽 ) ) )
15 9 14 mpbird ⊢ ( ( 𝜑 ∧ { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ) → ( 𝐵 ∈ ( 𝐼 ‘ 𝐽 ) ∧ 𝐴 ∈ ( 𝐼 ‘ 𝐽 ) ) )
16 15 simpld ⊢ ( ( 𝜑 ∧ { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ) → 𝐵 ∈ ( 𝐼 ‘ 𝐽 ) )
17 16 ex ⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) → 𝐵 ∈ ( 𝐼 ‘ 𝐽 ) ) )
18 simpr ⊢ ( ( 𝜑 ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) → { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) )
19 3 simp3d ⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
20 19 adantr ⊢ ( ( 𝜑 ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) → 𝐶 ∈ 𝑉 )
21 prssg ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( ( 𝐵 ∈ ( 𝐼 ‘ 𝐾 ) ∧ 𝐶 ∈ ( 𝐼 ‘ 𝐾 ) ) ↔ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
22 10 20 21 syl2an2r ⊢ ( ( 𝜑 ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) → ( ( 𝐵 ∈ ( 𝐼 ‘ 𝐾 ) ∧ 𝐶 ∈ ( 𝐼 ‘ 𝐾 ) ) ↔ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
23 18 22 mpbird ⊢ ( ( 𝜑 ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) → ( 𝐵 ∈ ( 𝐼 ‘ 𝐾 ) ∧ 𝐶 ∈ ( 𝐼 ‘ 𝐾 ) ) )
24 23 simpld ⊢ ( ( 𝜑 ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) → 𝐵 ∈ ( 𝐼 ‘ 𝐾 ) )
25 24 ex ⊢ ( 𝜑 → ( { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) → 𝐵 ∈ ( 𝐼 ‘ 𝐾 ) ) )
26 17 25 anim12d ⊢ ( 𝜑 → ( ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) → ( 𝐵 ∈ ( 𝐼 ‘ 𝐽 ) ∧ 𝐵 ∈ ( 𝐼 ‘ 𝐾 ) ) ) )
27 5 26 mpd ⊢ ( 𝜑 → ( 𝐵 ∈ ( 𝐼 ‘ 𝐽 ) ∧ 𝐵 ∈ ( 𝐼 ‘ 𝐾 ) ) )