Theorem f1oeq1

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

	Ref	Expression
Assertion	f1oeq1	$$⊢F=G\to \left(F:A\underset{1-1\; onto}{⟶}B\leftrightarrow G:A\underset{1-1\; onto}{⟶}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		f1eq1	$$⊢F=G\to \left(F:A\underset{1-1}{⟶}B\leftrightarrow G:A\underset{1-1}{⟶}B\right)$$
2		foeq1	$$⊢F=G\to \left(F:A\underset{onto}{⟶}B\leftrightarrow G:A\underset{onto}{⟶}B\right)$$
3	1 2	anbi12d	$$⊢F=G\to \left(\left(F:A\underset{1-1}{⟶}B\wedge F:A\underset{onto}{⟶}B\right)\leftrightarrow \left(G:A\underset{1-1}{⟶}B\wedge G:A\underset{onto}{⟶}B\right)\right)$$
4		df-f1o	$$⊢F:A\underset{1-1\; onto}{⟶}B\leftrightarrow \left(F:A\underset{1-1}{⟶}B\wedge F:A\underset{onto}{⟶}B\right)$$
5		df-f1o	$$⊢G:A\underset{1-1\; onto}{⟶}B\leftrightarrow \left(G:A\underset{1-1}{⟶}B\wedge G:A\underset{onto}{⟶}B\right)$$
6	3 4 5	3bitr4g	$$⊢F=G\to \left(F:A\underset{1-1\; onto}{⟶}B\leftrightarrow G:A\underset{1-1\; onto}{⟶}B\right)$$