Theorem ssimaexg

Description: The existence of a subimage. (Contributed by FL, 15-Apr-2007)

	Ref	Expression
Assertion	ssimaexg	⊢ ( ( 𝐴 ∈ 𝐶 ∧ Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝐴 ) ) → ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		imaeq2	⊢ ( 𝑦 = 𝐴 → ( 𝐹 “ 𝑦 ) = ( 𝐹 “ 𝐴 ) )
2	1	sseq2d	⊢ ( 𝑦 = 𝐴 → ( 𝐵 ⊆ ( 𝐹 “ 𝑦 ) ↔ 𝐵 ⊆ ( 𝐹 “ 𝐴 ) ) )
3	2	anbi2d	⊢ ( 𝑦 = 𝐴 → ( ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝑦 ) ) ↔ ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝐴 ) ) ) )
4		sseq2	⊢ ( 𝑦 = 𝐴 → ( 𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝐴 ) )
5	4	anbi1d	⊢ ( 𝑦 = 𝐴 → ( ( 𝑥 ⊆ 𝑦 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ↔ ( 𝑥 ⊆ 𝐴 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ) )
6	5	exbidv	⊢ ( 𝑦 = 𝐴 → ( ∃ 𝑥 ( 𝑥 ⊆ 𝑦 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ↔ ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ) )
7	3 6	imbi12d	⊢ ( 𝑦 = 𝐴 → ( ( ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝑦 ) ) → ∃ 𝑥 ( 𝑥 ⊆ 𝑦 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ) ↔ ( ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝐴 ) ) → ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ) ) )
8		vex	⊢ 𝑦 ∈ V
9	8	ssimaex	⊢ ( ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝑦 ) ) → ∃ 𝑥 ( 𝑥 ⊆ 𝑦 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) )
10	7 9	vtoclg	⊢ ( 𝐴 ∈ 𝐶 → ( ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝐴 ) ) → ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ) )
11	10	3impib	⊢ ( ( 𝐴 ∈ 𝐶 ∧ Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝐴 ) ) → ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) )