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Metamath Proof Explorer

Theorem fiss

Description: Subset relationship for function fi . (Contributed by Jeff Hankins, 7-Oct-2009) (Revised by Mario Carneiro, 24-Nov-2013)

Ref Expression
Assertion fiss ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ) → ( fi ‘ 𝐴 ) ⊆ ( fi ‘ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 sstr2 ⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐵 ⊆ 𝑦 → 𝐴 ⊆ 𝑦 ) )
2 1 adantl ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐵 ⊆ 𝑦 → 𝐴 ⊆ 𝑦 ) )
3 2 anim1d ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ) → ( ( 𝐵 ⊆ 𝑦 ∧ ∀ 𝑥 ∈ 𝑦 ∀ 𝑧 ∈ 𝑦 ( 𝑥 ∩ 𝑧 ) ∈ 𝑦 ) → ( 𝐴 ⊆ 𝑦 ∧ ∀ 𝑥 ∈ 𝑦 ∀ 𝑧 ∈ 𝑦 ( 𝑥 ∩ 𝑧 ) ∈ 𝑦 ) ) )
4 3 ss2abdv ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ) → { 𝑦 ∣ ( 𝐵 ⊆ 𝑦 ∧ ∀ 𝑥 ∈ 𝑦 ∀ 𝑧 ∈ 𝑦 ( 𝑥 ∩ 𝑧 ) ∈ 𝑦 ) } ⊆ { 𝑦 ∣ ( 𝐴 ⊆ 𝑦 ∧ ∀ 𝑥 ∈ 𝑦 ∀ 𝑧 ∈ 𝑦 ( 𝑥 ∩ 𝑧 ) ∈ 𝑦 ) } )
5 intss ⊢ ( { 𝑦 ∣ ( 𝐵 ⊆ 𝑦 ∧ ∀ 𝑥 ∈ 𝑦 ∀ 𝑧 ∈ 𝑦 ( 𝑥 ∩ 𝑧 ) ∈ 𝑦 ) } ⊆ { 𝑦 ∣ ( 𝐴 ⊆ 𝑦 ∧ ∀ 𝑥 ∈ 𝑦 ∀ 𝑧 ∈ 𝑦 ( 𝑥 ∩ 𝑧 ) ∈ 𝑦 ) } → ∩ { 𝑦 ∣ ( 𝐴 ⊆ 𝑦 ∧ ∀ 𝑥 ∈ 𝑦 ∀ 𝑧 ∈ 𝑦 ( 𝑥 ∩ 𝑧 ) ∈ 𝑦 ) } ⊆ ∩ { 𝑦 ∣ ( 𝐵 ⊆ 𝑦 ∧ ∀ 𝑥 ∈ 𝑦 ∀ 𝑧 ∈ 𝑦 ( 𝑥 ∩ 𝑧 ) ∈ 𝑦 ) } )
6 4 5 syl ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ) → ∩ { 𝑦 ∣ ( 𝐴 ⊆ 𝑦 ∧ ∀ 𝑥 ∈ 𝑦 ∀ 𝑧 ∈ 𝑦 ( 𝑥 ∩ 𝑧 ) ∈ 𝑦 ) } ⊆ ∩ { 𝑦 ∣ ( 𝐵 ⊆ 𝑦 ∧ ∀ 𝑥 ∈ 𝑦 ∀ 𝑧 ∈ 𝑦 ( 𝑥 ∩ 𝑧 ) ∈ 𝑦 ) } )
7 ssexg ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ) → 𝐴 ∈ V )
8 7 ancoms ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ∈ V )
9 dffi2 ⊢ ( 𝐴 ∈ V → ( fi ‘ 𝐴 ) = ∩ { 𝑦 ∣ ( 𝐴 ⊆ 𝑦 ∧ ∀ 𝑥 ∈ 𝑦 ∀ 𝑧 ∈ 𝑦 ( 𝑥 ∩ 𝑧 ) ∈ 𝑦 ) } )
10 8 9 syl ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ) → ( fi ‘ 𝐴 ) = ∩ { 𝑦 ∣ ( 𝐴 ⊆ 𝑦 ∧ ∀ 𝑥 ∈ 𝑦 ∀ 𝑧 ∈ 𝑦 ( 𝑥 ∩ 𝑧 ) ∈ 𝑦 ) } )
11 dffi2 ⊢ ( 𝐵 ∈ 𝑉 → ( fi ‘ 𝐵 ) = ∩ { 𝑦 ∣ ( 𝐵 ⊆ 𝑦 ∧ ∀ 𝑥 ∈ 𝑦 ∀ 𝑧 ∈ 𝑦 ( 𝑥 ∩ 𝑧 ) ∈ 𝑦 ) } )
12 11 adantr ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ) → ( fi ‘ 𝐵 ) = ∩ { 𝑦 ∣ ( 𝐵 ⊆ 𝑦 ∧ ∀ 𝑥 ∈ 𝑦 ∀ 𝑧 ∈ 𝑦 ( 𝑥 ∩ 𝑧 ) ∈ 𝑦 ) } )
13 6 10 12 3sstr4d ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵 ) → ( fi ‘ 𝐴 ) ⊆ ( fi ‘ 𝐵 ) )