Theorem fnbrafv2b

Description: Equivalence of function value and binary relation, analogous to fnbrfvb . (Contributed by AV, 6-Sep-2022)

	Ref	Expression
Assertion	fnbrafv2b	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( ( 𝐹 '''' 𝐵 ) = 𝐶 ↔ 𝐵 𝐹 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( 𝐹 '''' 𝐵 ) = ( 𝐹 '''' 𝐵 )
2		fundmdfat	⊢ ( ( Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹 ) → 𝐹 defAt 𝐵 )
3	2	funfni	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → 𝐹 defAt 𝐵 )
4		dfatafv2ex	⊢ ( 𝐹 defAt 𝐵 → ( 𝐹 '''' 𝐵 ) ∈ V )
5	3 4	syl	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( 𝐹 '''' 𝐵 ) ∈ V )
6		eqeq2	⊢ ( 𝑥 = ( 𝐹 '''' 𝐵 ) → ( ( 𝐹 '''' 𝐵 ) = 𝑥 ↔ ( 𝐹 '''' 𝐵 ) = ( 𝐹 '''' 𝐵 ) ) )
7		breq2	⊢ ( 𝑥 = ( 𝐹 '''' 𝐵 ) → ( 𝐵 𝐹 𝑥 ↔ 𝐵 𝐹 ( 𝐹 '''' 𝐵 ) ) )
8	6 7	bibi12d	⊢ ( 𝑥 = ( 𝐹 '''' 𝐵 ) → ( ( ( 𝐹 '''' 𝐵 ) = 𝑥 ↔ 𝐵 𝐹 𝑥 ) ↔ ( ( 𝐹 '''' 𝐵 ) = ( 𝐹 '''' 𝐵 ) ↔ 𝐵 𝐹 ( 𝐹 '''' 𝐵 ) ) ) )
9	8	adantl	⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) ∧ 𝑥 = ( 𝐹 '''' 𝐵 ) ) → ( ( ( 𝐹 '''' 𝐵 ) = 𝑥 ↔ 𝐵 𝐹 𝑥 ) ↔ ( ( 𝐹 '''' 𝐵 ) = ( 𝐹 '''' 𝐵 ) ↔ 𝐵 𝐹 ( 𝐹 '''' 𝐵 ) ) ) )
10		fneu	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ∃! 𝑥 𝐵 𝐹 𝑥 )
11		tz6.12c-afv2	⊢ ( ∃! 𝑥 𝐵 𝐹 𝑥 → ( ( 𝐹 '''' 𝐵 ) = 𝑥 ↔ 𝐵 𝐹 𝑥 ) )
12	10 11	syl	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( ( 𝐹 '''' 𝐵 ) = 𝑥 ↔ 𝐵 𝐹 𝑥 ) )
13	5 9 12	vtocld	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( ( 𝐹 '''' 𝐵 ) = ( 𝐹 '''' 𝐵 ) ↔ 𝐵 𝐹 ( 𝐹 '''' 𝐵 ) ) )
14	1 13	mpbii	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → 𝐵 𝐹 ( 𝐹 '''' 𝐵 ) )
15		breq2	⊢ ( ( 𝐹 '''' 𝐵 ) = 𝐶 → ( 𝐵 𝐹 ( 𝐹 '''' 𝐵 ) ↔ 𝐵 𝐹 𝐶 ) )
16	14 15	syl5ibcom	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( ( 𝐹 '''' 𝐵 ) = 𝐶 → 𝐵 𝐹 𝐶 ) )
17		fnfun	⊢ ( 𝐹 Fn 𝐴 → Fun 𝐹 )
18		funbrafv2	⊢ ( Fun 𝐹 → ( 𝐵 𝐹 𝐶 → ( 𝐹 '''' 𝐵 ) = 𝐶 ) )
19	17 18	syl	⊢ ( 𝐹 Fn 𝐴 → ( 𝐵 𝐹 𝐶 → ( 𝐹 '''' 𝐵 ) = 𝐶 ) )
20	19	adantr	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( 𝐵 𝐹 𝐶 → ( 𝐹 '''' 𝐵 ) = 𝐶 ) )
21	16 20	impbid	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( ( 𝐹 '''' 𝐵 ) = 𝐶 ↔ 𝐵 𝐹 𝐶 ) )