Theorem k0004lem1

Description: Application of ssin to range of a function. (Contributed by RP, 1-Apr-2021)

	Ref	Expression
Assertion	k0004lem1	⊢ ( 𝐷 = ( 𝐵 ∩ 𝐶 ) → ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ) ↔ 𝐹 : 𝐴 ⟶ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		fnima	⊢ ( 𝐹 Fn 𝐴 → ( 𝐹 “ 𝐴 ) = ran 𝐹 )
2	1	sseq1d	⊢ ( 𝐹 Fn 𝐴 → ( ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ↔ ran 𝐹 ⊆ 𝐶 ) )
3	2	anbi2d	⊢ ( 𝐹 Fn 𝐴 → ( ( ran 𝐹 ⊆ 𝐵 ∧ ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ) ↔ ( ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ⊆ 𝐶 ) ) )
4		ssin	⊢ ( ( ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ⊆ 𝐶 ) ↔ ran 𝐹 ⊆ ( 𝐵 ∩ 𝐶 ) )
5	3 4	bitrdi	⊢ ( 𝐹 Fn 𝐴 → ( ( ran 𝐹 ⊆ 𝐵 ∧ ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ) ↔ ran 𝐹 ⊆ ( 𝐵 ∩ 𝐶 ) ) )
6	5	pm5.32i	⊢ ( ( 𝐹 Fn 𝐴 ∧ ( ran 𝐹 ⊆ 𝐵 ∧ ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ) ) ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ( 𝐵 ∩ 𝐶 ) ) )
7		df-f	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) )
8	7	anbi1i	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ) ↔ ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) ∧ ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ) )
9		anass	⊢ ( ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) ∧ ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ) ↔ ( 𝐹 Fn 𝐴 ∧ ( ran 𝐹 ⊆ 𝐵 ∧ ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ) ) )
10	8 9	bitri	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ) ↔ ( 𝐹 Fn 𝐴 ∧ ( ran 𝐹 ⊆ 𝐵 ∧ ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ) ) )
11		df-f	⊢ ( 𝐹 : 𝐴 ⟶ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ ( 𝐵 ∩ 𝐶 ) ) )
12	6 10 11	3bitr4i	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ) ↔ 𝐹 : 𝐴 ⟶ ( 𝐵 ∩ 𝐶 ) )
13		feq3	⊢ ( 𝐷 = ( 𝐵 ∩ 𝐶 ) → ( 𝐹 : 𝐴 ⟶ 𝐷 ↔ 𝐹 : 𝐴 ⟶ ( 𝐵 ∩ 𝐶 ) ) )
14	12 13	bitr4id	⊢ ( 𝐷 = ( 𝐵 ∩ 𝐶 ) → ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ) ↔ 𝐹 : 𝐴 ⟶ 𝐷 ) )