Theorem elsetpreimafveqfv

Description: The elements of the preimage of a function value have the same function values. (Contributed by AV, 5-Mar-2024)

	Ref	Expression
Hypothesis	setpreimafvex.p	⊢ 𝑃 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) }
Assertion	elsetpreimafveqfv	⊢ ( ( 𝐹 Fn 𝐴 ∧ ( 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) ) → ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		setpreimafvex.p	⊢ 𝑃 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) }
2	1	elsetpreimafvbi	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆 ) → ( 𝑌 ∈ 𝑆 ↔ ( 𝑌 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑌 ) = ( 𝐹 ‘ 𝑋 ) ) ) )
3		simpr	⊢ ( ( 𝑌 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑌 ) = ( 𝐹 ‘ 𝑋 ) ) → ( 𝐹 ‘ 𝑌 ) = ( 𝐹 ‘ 𝑋 ) )
4	3	eqcomd	⊢ ( ( 𝑌 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑌 ) = ( 𝐹 ‘ 𝑋 ) ) → ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑌 ) )
5	2 4	biimtrdi	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆 ) → ( 𝑌 ∈ 𝑆 → ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑌 ) ) )
6	5	3exp	⊢ ( 𝐹 Fn 𝐴 → ( 𝑆 ∈ 𝑃 → ( 𝑋 ∈ 𝑆 → ( 𝑌 ∈ 𝑆 → ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑌 ) ) ) ) )
7	6	3imp2	⊢ ( ( 𝐹 Fn 𝐴 ∧ ( 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) ) → ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑌 ) )