Theorem brfullfun

Description: A binary relation form condition for the full function. (Contributed by Scott Fenton, 17-Apr-2014) (Revised by Mario Carneiro, 19-Apr-2014)

	Ref	Expression
Hypotheses	brfullfun.1	⊢ 𝐴 ∈ V
	brfullfun.2	⊢ 𝐵 ∈ V
Assertion	brfullfun	⊢ ( 𝐴 FullFun 𝐹 𝐵 ↔ 𝐵 = ( 𝐹 ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		brfullfun.1	⊢ 𝐴 ∈ V
2		brfullfun.2	⊢ 𝐵 ∈ V
3		eqcom	⊢ ( ( FullFun 𝐹 ‘ 𝐴 ) = 𝐵 ↔ 𝐵 = ( FullFun 𝐹 ‘ 𝐴 ) )
4		fullfunfnv	⊢ FullFun 𝐹 Fn V
5		fnbrfvb	⊢ ( ( FullFun 𝐹 Fn V ∧ 𝐴 ∈ V ) → ( ( FullFun 𝐹 ‘ 𝐴 ) = 𝐵 ↔ 𝐴 FullFun 𝐹 𝐵 ) )
6	4 1 5	mp2an	⊢ ( ( FullFun 𝐹 ‘ 𝐴 ) = 𝐵 ↔ 𝐴 FullFun 𝐹 𝐵 )
7		fullfunfv	⊢ ( FullFun 𝐹 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 )
8	7	eqeq2i	⊢ ( 𝐵 = ( FullFun 𝐹 ‘ 𝐴 ) ↔ 𝐵 = ( 𝐹 ‘ 𝐴 ) )
9	3 6 8	3bitr3i	⊢ ( 𝐴 FullFun 𝐹 𝐵 ↔ 𝐵 = ( 𝐹 ‘ 𝐴 ) )