Theorem dfse3

Description: Alternate definition of set-like relationships. (Contributed by Scott Fenton, 19-Aug-2024)

	Ref	Expression
Assertion	dfse3	⊢ ( 𝑅 Se 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 Pred ( 𝑅 , 𝐴 , 𝑥 ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		dfse2	⊢ ( 𝑅 Se 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 ( 𝐴 ∩ ( ◡ 𝑅 “ { 𝑥 } ) ) ∈ V )
2		df-pred	⊢ Pred ( 𝑅 , 𝐴 , 𝑥 ) = ( 𝐴 ∩ ( ◡ 𝑅 “ { 𝑥 } ) )
3	2	eleq1i	⊢ ( Pred ( 𝑅 , 𝐴 , 𝑥 ) ∈ V ↔ ( 𝐴 ∩ ( ◡ 𝑅 “ { 𝑥 } ) ) ∈ V )
4	3	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 Pred ( 𝑅 , 𝐴 , 𝑥 ) ∈ V ↔ ∀ 𝑥 ∈ 𝐴 ( 𝐴 ∩ ( ◡ 𝑅 “ { 𝑥 } ) ) ∈ V )
5	1 4	bitr4i	⊢ ( 𝑅 Se 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 Pred ( 𝑅 , 𝐴 , 𝑥 ) ∈ V )