Theorem f1oeq2

Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997)

	Ref	Expression
Assertion	f1oeq2	$$⊢A=B\to \left(F:A\underset{1-1\; onto}{⟶}C\leftrightarrow F:B\underset{1-1\; onto}{⟶}C\right)$$

Proof

Step	Hyp	Ref	Expression
1		f1eq2	$$⊢A=B\to \left(F:A\underset{1-1}{⟶}C\leftrightarrow F:B\underset{1-1}{⟶}C\right)$$
2		foeq2	$$⊢A=B\to \left(F:A\underset{onto}{⟶}C\leftrightarrow F:B\underset{onto}{⟶}C\right)$$
3	1 2	anbi12d	$$⊢A=B\to \left(\left(F:A\underset{1-1}{⟶}C\wedge F:A\underset{onto}{⟶}C\right)\leftrightarrow \left(F:B\underset{1-1}{⟶}C\wedge F:B\underset{onto}{⟶}C\right)\right)$$
4		df-f1o	$$⊢F:A\underset{1-1\; onto}{⟶}C\leftrightarrow \left(F:A\underset{1-1}{⟶}C\wedge F:A\underset{onto}{⟶}C\right)$$
5		df-f1o	$$⊢F:B\underset{1-1\; onto}{⟶}C\leftrightarrow \left(F:B\underset{1-1}{⟶}C\wedge F:B\underset{onto}{⟶}C\right)$$
6	3 4 5	3bitr4g	$$⊢A=B\to \left(F:A\underset{1-1\; onto}{⟶}C\leftrightarrow F:B\underset{1-1\; onto}{⟶}C\right)$$