Theorem f1oeq23

Description: Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012)

	Ref	Expression
Assertion	f1oeq23	$$⊢\left(A=B\wedge C=D\right)\to \left(F:A\underset{1-1\; onto}{⟶}C\leftrightarrow F:B\underset{1-1\; onto}{⟶}D\right)$$

Proof

Step	Hyp	Ref	Expression
1		f1oeq2	$$⊢A=B\to \left(F:A\underset{1-1\; onto}{⟶}C\leftrightarrow F:B\underset{1-1\; onto}{⟶}C\right)$$
2		f1oeq3	$$⊢C=D\to \left(F:B\underset{1-1\; onto}{⟶}C\leftrightarrow F:B\underset{1-1\; onto}{⟶}D\right)$$
3	1 2	sylan9bb	$$⊢\left(A=B\wedge C=D\right)\to \left(F:A\underset{1-1\; onto}{⟶}C\leftrightarrow F:B\underset{1-1\; onto}{⟶}D\right)$$