Theorem feq3

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

	Ref	Expression
Assertion	feq3	$$⊢A=B\to \left(F:C⟶A\leftrightarrow F:C⟶B\right)$$

Proof

Step	Hyp	Ref	Expression
1		sseq2	$$⊢A=B\to \left(\mathrm{ran}F\subseteq A\leftrightarrow \mathrm{ran}F\subseteq B\right)$$
2	1	anbi2d	$$⊢A=B\to \left(\left(FFnC\wedge \mathrm{ran}F\subseteq A\right)\leftrightarrow \left(FFnC\wedge \mathrm{ran}F\subseteq B\right)\right)$$
3		df-f	$$⊢F:C⟶A\leftrightarrow \left(FFnC\wedge \mathrm{ran}F\subseteq A\right)$$
4		df-f	$$⊢F:C⟶B\leftrightarrow \left(FFnC\wedge \mathrm{ran}F\subseteq B\right)$$
5	2 3 4	3bitr4g	$$⊢A=B\to \left(F:C⟶A\leftrightarrow F:C⟶B\right)$$