Theorem f1oeq2d

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

	Ref	Expression
Hypothesis	f1oeq2d.1	$$⊢\phi \to A=B$$
Assertion	f1oeq2d	$$⊢\phi \to \left(F:A\underset{1-1\; onto}{⟶}C\leftrightarrow F:B\underset{1-1\; onto}{⟶}C\right)$$

Proof

Step	Hyp	Ref	Expression
1		f1oeq2d.1	$$⊢\phi \to A=B$$
2		f1oeq2	$$⊢A=B\to \left(F:A\underset{1-1\; onto}{⟶}C\leftrightarrow F:B\underset{1-1\; onto}{⟶}C\right)$$
3	1 2	syl	$$⊢\phi \to \left(F:A\underset{1-1\; onto}{⟶}C\leftrightarrow F:B\underset{1-1\; onto}{⟶}C\right)$$