Theorem rankfu

Description: An upper bound on the rank of a function. (Contributed by Gérard Lang, 5-Aug-2018)

	Ref	Expression
Hypotheses	rankxpl.1	$$⊢A\in \mathrm{V}$$
	rankxpl.2	$$⊢B\in \mathrm{V}$$
Assertion	rankfu	$$⊢F:A⟶B\to \mathrm{rank}\left(F\right)\subseteq \mathrm{suc}\mathrm{suc}\mathrm{rank}\left(A\cup B\right)$$

Proof

Step	Hyp	Ref	Expression
1		rankxpl.1	$$⊢A\in \mathrm{V}$$
2		rankxpl.2	$$⊢B\in \mathrm{V}$$
3		fssxp	$$⊢F:A⟶B\to F\subseteq A\times B$$
4	1 2	xpex	$$⊢A\times B\in \mathrm{V}$$
5	4	rankss	$$⊢F\subseteq A\times B\to \mathrm{rank}\left(F\right)\subseteq \mathrm{rank}\left(A\times B\right)$$
6	1 2	rankxpu	$$⊢\mathrm{rank}\left(A\times B\right)\subseteq \mathrm{suc}\mathrm{suc}\mathrm{rank}\left(A\cup B\right)$$
7	5 6	sstrdi	$$⊢F\subseteq A\times B\to \mathrm{rank}\left(F\right)\subseteq \mathrm{suc}\mathrm{suc}\mathrm{rank}\left(A\cup B\right)$$
8	3 7	syl	$$⊢F:A⟶B\to \mathrm{rank}\left(F\right)\subseteq \mathrm{suc}\mathrm{suc}\mathrm{rank}\left(A\cup B\right)$$