Theorem inisegn0a

Description: The inverse image of a singleton subset of an image is non-empty. (Contributed by Zhi Wang, 7-Nov-2025)

	Ref	Expression
Assertion	inisegn0a	⊢ ( 𝐴 ∈ ( 𝐹 “ 𝐵 ) → ( ◡ 𝐹 “ { 𝐴 } ) ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		elimag	⊢ ( 𝐴 ∈ ( 𝐹 “ 𝐵 ) → ( 𝐴 ∈ ( 𝐹 “ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐵 𝑥 𝐹 𝐴 ) )
2	1	ibi	⊢ ( 𝐴 ∈ ( 𝐹 “ 𝐵 ) → ∃ 𝑥 ∈ 𝐵 𝑥 𝐹 𝐴 )
3		vex	⊢ 𝑥 ∈ V
4	3	eliniseg	⊢ ( 𝐴 ∈ ( 𝐹 “ 𝐵 ) → ( 𝑥 ∈ ( ◡ 𝐹 “ { 𝐴 } ) ↔ 𝑥 𝐹 𝐴 ) )
5		ne0i	⊢ ( 𝑥 ∈ ( ◡ 𝐹 “ { 𝐴 } ) → ( ◡ 𝐹 “ { 𝐴 } ) ≠ ∅ )
6	4 5	biimtrrdi	⊢ ( 𝐴 ∈ ( 𝐹 “ 𝐵 ) → ( 𝑥 𝐹 𝐴 → ( ◡ 𝐹 “ { 𝐴 } ) ≠ ∅ ) )
7	6	rexlimdvw	⊢ ( 𝐴 ∈ ( 𝐹 “ 𝐵 ) → ( ∃ 𝑥 ∈ 𝐵 𝑥 𝐹 𝐴 → ( ◡ 𝐹 “ { 𝐴 } ) ≠ ∅ ) )
8	2 7	mpd	⊢ ( 𝐴 ∈ ( 𝐹 “ 𝐵 ) → ( ◡ 𝐹 “ { 𝐴 } ) ≠ ∅ )