Metamath Proof Explorer


Theorem ttukeylem2

Description: Lemma for ttukey . A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015)

Ref Expression
Hypotheses ttukeylem.1 ( 𝜑𝐹 : ( card ‘ ( 𝐴𝐵 ) ) –1-1-onto→ ( 𝐴𝐵 ) )
ttukeylem.2 ( 𝜑𝐵𝐴 )
ttukeylem.3 ( 𝜑 → ∀ 𝑥 ( 𝑥𝐴 ↔ ( 𝒫 𝑥 ∩ Fin ) ⊆ 𝐴 ) )
Assertion ttukeylem2 ( ( 𝜑 ∧ ( 𝐶𝐴𝐷𝐶 ) ) → 𝐷𝐴 )

Proof

Step Hyp Ref Expression
1 ttukeylem.1 ( 𝜑𝐹 : ( card ‘ ( 𝐴𝐵 ) ) –1-1-onto→ ( 𝐴𝐵 ) )
2 ttukeylem.2 ( 𝜑𝐵𝐴 )
3 ttukeylem.3 ( 𝜑 → ∀ 𝑥 ( 𝑥𝐴 ↔ ( 𝒫 𝑥 ∩ Fin ) ⊆ 𝐴 ) )
4 simpr ( ( 𝜑𝐷𝐶 ) → 𝐷𝐶 )
5 4 sspwd ( ( 𝜑𝐷𝐶 ) → 𝒫 𝐷 ⊆ 𝒫 𝐶 )
6 ssrin ( 𝒫 𝐷 ⊆ 𝒫 𝐶 → ( 𝒫 𝐷 ∩ Fin ) ⊆ ( 𝒫 𝐶 ∩ Fin ) )
7 sstr2 ( ( 𝒫 𝐷 ∩ Fin ) ⊆ ( 𝒫 𝐶 ∩ Fin ) → ( ( 𝒫 𝐶 ∩ Fin ) ⊆ 𝐴 → ( 𝒫 𝐷 ∩ Fin ) ⊆ 𝐴 ) )
8 5 6 7 3syl ( ( 𝜑𝐷𝐶 ) → ( ( 𝒫 𝐶 ∩ Fin ) ⊆ 𝐴 → ( 𝒫 𝐷 ∩ Fin ) ⊆ 𝐴 ) )
9 1 2 3 ttukeylem1 ( 𝜑 → ( 𝐶𝐴 ↔ ( 𝒫 𝐶 ∩ Fin ) ⊆ 𝐴 ) )
10 9 adantr ( ( 𝜑𝐷𝐶 ) → ( 𝐶𝐴 ↔ ( 𝒫 𝐶 ∩ Fin ) ⊆ 𝐴 ) )
11 1 2 3 ttukeylem1 ( 𝜑 → ( 𝐷𝐴 ↔ ( 𝒫 𝐷 ∩ Fin ) ⊆ 𝐴 ) )
12 11 adantr ( ( 𝜑𝐷𝐶 ) → ( 𝐷𝐴 ↔ ( 𝒫 𝐷 ∩ Fin ) ⊆ 𝐴 ) )
13 8 10 12 3imtr4d ( ( 𝜑𝐷𝐶 ) → ( 𝐶𝐴𝐷𝐴 ) )
14 13 impancom ( ( 𝜑𝐶𝐴 ) → ( 𝐷𝐶𝐷𝐴 ) )
15 14 impr ( ( 𝜑 ∧ ( 𝐶𝐴𝐷𝐶 ) ) → 𝐷𝐴 )