Theorem foeq3

Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994)

	Ref	Expression
Assertion	foeq3	$$⊢A=B\to \left(F:C\underset{onto}{⟶}A\leftrightarrow F:C\underset{onto}{⟶}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqeq2	$$⊢A=B\to \left(\mathrm{ran}F=A\leftrightarrow \mathrm{ran}F=B\right)$$
2	1	anbi2d	$$⊢A=B\to \left(\left(FFnC\wedge \mathrm{ran}F=A\right)\leftrightarrow \left(FFnC\wedge \mathrm{ran}F=B\right)\right)$$
3		df-fo	$$⊢F:C\underset{onto}{⟶}A\leftrightarrow \left(FFnC\wedge \mathrm{ran}F=A\right)$$
4		df-fo	$$⊢F:C\underset{onto}{⟶}B\leftrightarrow \left(FFnC\wedge \mathrm{ran}F=B\right)$$
5	2 3 4	3bitr4g	$$⊢A=B\to \left(F:C\underset{onto}{⟶}A\leftrightarrow F:C\underset{onto}{⟶}B\right)$$