Theorem f1oeq3d

Description: Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypothesis	f1oeq3d.1	$$⊢\phi \to A=B$$
Assertion	f1oeq3d	$$⊢\phi \to \left(F:C\underset{1-1\; onto}{⟶}A\leftrightarrow F:C\underset{1-1\; onto}{⟶}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		f1oeq3d.1	$$⊢\phi \to A=B$$
2		f1oeq3	$$⊢A=B\to \left(F:C\underset{1-1\; onto}{⟶}A\leftrightarrow F:C\underset{1-1\; onto}{⟶}B\right)$$
3	1 2	syl	$$⊢\phi \to \left(F:C\underset{1-1\; onto}{⟶}A\leftrightarrow F:C\underset{1-1\; onto}{⟶}B\right)$$