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 ) )