Metamath Proof Explorer


Theorem xpfir

Description: The components of a nonempty finite Cartesian product are finite. (Contributed by Paul Chapman, 11-Apr-2009) (Proof shortened by Mario Carneiro, 29-Apr-2015)

Ref Expression
Assertion xpfir A × B Fin A × B A Fin B Fin

Proof

Step Hyp Ref Expression
1 xpexr2 A × B Fin A × B A V B V
2 1 simpld A × B Fin A × B A V
3 1 simprd A × B Fin A × B B V
4 simpr A × B Fin A × B A × B
5 xpnz A B A × B
6 4 5 sylibr A × B Fin A × B A B
7 6 simprd A × B Fin A × B B
8 xpdom3 A V B V B A A × B
9 2 3 7 8 syl3anc A × B Fin A × B A A × B
10 domfi A × B Fin A A × B A Fin
11 9 10 syldan A × B Fin A × B A Fin
12 6 simpld A × B Fin A × B A
13 xpdom3 B V A V A B B × A
14 3 2 12 13 syl3anc A × B Fin A × B B B × A
15 xpcomeng B V A V B × A A × B
16 3 2 15 syl2anc A × B Fin A × B B × A A × B
17 domentr B B × A B × A A × B B A × B
18 14 16 17 syl2anc A × B Fin A × B B A × B
19 domfi A × B Fin B A × B B Fin
20 18 19 syldan A × B Fin A × B B Fin
21 11 20 jca A × B Fin A × B A Fin B Fin