Theorem f1domg

Description: The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004)

	Ref	Expression
Assertion	f1domg	$$⊢B\in C\to \left(F:A\underset{1-1}{⟶}B\to A\preccurlyeq B\right)$$

Proof

Step	Hyp	Ref	Expression
1		f1f	$$⊢F:A\underset{1-1}{⟶}B\to F:A⟶B$$
2		f1dmex	$$⊢\left(F:A\underset{1-1}{⟶}B\wedge B\in C\right)\to A\in \mathrm{V}$$
3		fex	$$⊢\left(F:A⟶B\wedge A\in \mathrm{V}\right)\to F\in \mathrm{V}$$
4	1 2 3	syl2an2r	$$⊢\left(F:A\underset{1-1}{⟶}B\wedge B\in C\right)\to F\in \mathrm{V}$$
5	4	expcom	$$⊢B\in C\to \left(F:A\underset{1-1}{⟶}B\to F\in \mathrm{V}\right)$$
6		f1eq1	$$⊢f=F\to \left(f:A\underset{1-1}{⟶}B\leftrightarrow F:A\underset{1-1}{⟶}B\right)$$
7	6	spcegv	$$⊢F\in \mathrm{V}\to \left(F:A\underset{1-1}{⟶}B\to \exists ff:A\underset{1-1}{⟶}B\right)$$
8	5 7	syli	$$⊢B\in C\to \left(F:A\underset{1-1}{⟶}B\to \exists ff:A\underset{1-1}{⟶}B\right)$$
9		brdomg	$$⊢B\in C\to \left(A\preccurlyeq B\leftrightarrow \exists ff:A\underset{1-1}{⟶}B\right)$$
10	8 9	sylibrd	$$⊢B\in C\to \left(F:A\underset{1-1}{⟶}B\to A\preccurlyeq B\right)$$