Theorem f1oeq3dd

Description: Equality deduction for one-to-one onto functions. (Contributed by Thierry Arnoux, 10-Jan-2026)

	Ref	Expression
Hypotheses	f1oeq3dd.1	⊢ ( 𝜑 → 𝐹 : 𝐶 –1-1-onto→ 𝐴 )
	f1oeq3dd.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	f1oeq3dd	⊢ ( 𝜑 → 𝐹 : 𝐶 –1-1-onto→ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		f1oeq3dd.1	⊢ ( 𝜑 → 𝐹 : 𝐶 –1-1-onto→ 𝐴 )
2		f1oeq3dd.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
3	2	f1oeq3d	⊢ ( 𝜑 → ( 𝐹 : 𝐶 –1-1-onto→ 𝐴 ↔ 𝐹 : 𝐶 –1-1-onto→ 𝐵 ) )
4	1 3	mpbid	⊢ ( 𝜑 → 𝐹 : 𝐶 –1-1-onto→ 𝐵 )