Theorem elimasni

Description: Membership in an image of a singleton. (Contributed by NM, 5-Aug-2010)

	Ref	Expression
Assertion	elimasni	⊢ ( 𝐶 ∈ ( 𝐴 “ { 𝐵 } ) → 𝐵 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		noel	⊢ ¬ 𝐶 ∈ ∅
2		snprc	⊢ ( ¬ 𝐵 ∈ V ↔ { 𝐵 } = ∅ )
3	2	biimpi	⊢ ( ¬ 𝐵 ∈ V → { 𝐵 } = ∅ )
4	3	imaeq2d	⊢ ( ¬ 𝐵 ∈ V → ( 𝐴 “ { 𝐵 } ) = ( 𝐴 “ ∅ ) )
5		ima0	⊢ ( 𝐴 “ ∅ ) = ∅
6	4 5	eqtrdi	⊢ ( ¬ 𝐵 ∈ V → ( 𝐴 “ { 𝐵 } ) = ∅ )
7	6	eleq2d	⊢ ( ¬ 𝐵 ∈ V → ( 𝐶 ∈ ( 𝐴 “ { 𝐵 } ) ↔ 𝐶 ∈ ∅ ) )
8	1 7	mtbiri	⊢ ( ¬ 𝐵 ∈ V → ¬ 𝐶 ∈ ( 𝐴 “ { 𝐵 } ) )
9	8	con4i	⊢ ( 𝐶 ∈ ( 𝐴 “ { 𝐵 } ) → 𝐵 ∈ V )
10		elex	⊢ ( 𝐶 ∈ ( 𝐴 “ { 𝐵 } ) → 𝐶 ∈ V )
11	9 10	jca	⊢ ( 𝐶 ∈ ( 𝐴 “ { 𝐵 } ) → ( 𝐵 ∈ V ∧ 𝐶 ∈ V ) )
12		elimasng	⊢ ( ( 𝐵 ∈ V ∧ 𝐶 ∈ V ) → ( 𝐶 ∈ ( 𝐴 “ { 𝐵 } ) ↔ ⟨ 𝐵 , 𝐶 ⟩ ∈ 𝐴 ) )
13		df-br	⊢ ( 𝐵 𝐴 𝐶 ↔ ⟨ 𝐵 , 𝐶 ⟩ ∈ 𝐴 )
14	12 13	bitr4di	⊢ ( ( 𝐵 ∈ V ∧ 𝐶 ∈ V ) → ( 𝐶 ∈ ( 𝐴 “ { 𝐵 } ) ↔ 𝐵 𝐴 𝐶 ) )
15	14	biimpd	⊢ ( ( 𝐵 ∈ V ∧ 𝐶 ∈ V ) → ( 𝐶 ∈ ( 𝐴 “ { 𝐵 } ) → 𝐵 𝐴 𝐶 ) )
16	11 15	mpcom	⊢ ( 𝐶 ∈ ( 𝐴 “ { 𝐵 } ) → 𝐵 𝐴 𝐶 )