Theorem imaeqsalv

Description: Substitute a function value into a universal quantifier over an image. (Contributed by Scott Fenton, 27-Sep-2024)

	Ref	Expression
Hypothesis	imaeqsex.1	⊢ ( 𝑥 = ( 𝐹 ‘ 𝑦 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion	imaeqsalv	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( ∀ 𝑥 ∈ ( 𝐹 “ 𝐵 ) 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		imaeqsex.1	⊢ ( 𝑥 = ( 𝐹 ‘ 𝑦 ) → ( 𝜑 ↔ 𝜓 ) )
2	1	notbid	⊢ ( 𝑥 = ( 𝐹 ‘ 𝑦 ) → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
3	2	imaeqsexv	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( ∃ 𝑥 ∈ ( 𝐹 “ 𝐵 ) ¬ 𝜑 ↔ ∃ 𝑦 ∈ 𝐵 ¬ 𝜓 ) )
4	3	notbid	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( ¬ ∃ 𝑥 ∈ ( 𝐹 “ 𝐵 ) ¬ 𝜑 ↔ ¬ ∃ 𝑦 ∈ 𝐵 ¬ 𝜓 ) )
5		dfral2	⊢ ( ∀ 𝑥 ∈ ( 𝐹 “ 𝐵 ) 𝜑 ↔ ¬ ∃ 𝑥 ∈ ( 𝐹 “ 𝐵 ) ¬ 𝜑 )
6		dfral2	⊢ ( ∀ 𝑦 ∈ 𝐵 𝜓 ↔ ¬ ∃ 𝑦 ∈ 𝐵 ¬ 𝜓 )
7	4 5 6	3bitr4g	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( ∀ 𝑥 ∈ ( 𝐹 “ 𝐵 ) 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 𝜓 ) )