Metamath Proof Explorer


Theorem dprdub

Description: Each factor is a subset of the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016)

Ref Expression
Hypotheses dprdub.1 φ G dom DProd S
dprdub.2 φ dom S = I
dprdub.3 φ X I
Assertion dprdub φ S X G DProd S

Proof

Step Hyp Ref Expression
1 dprdub.1 φ G dom DProd S
2 dprdub.2 φ dom S = I
3 dprdub.3 φ X I
4 eqid 0 G = 0 G
5 eqid h i I S i | finSupp 0 G h = h i I S i | finSupp 0 G h
6 1 adantr φ x S X G dom DProd S
7 2 adantr φ x S X dom S = I
8 3 adantr φ x S X X I
9 simpr φ x S X x S X
10 eqid n I if n = X x 0 G = n I if n = X x 0 G
11 4 5 6 7 8 9 10 dprdfid φ x S X n I if n = X x 0 G h i I S i | finSupp 0 G h G n I if n = X x 0 G = x
12 11 simprd φ x S X G n I if n = X x 0 G = x
13 11 simpld φ x S X n I if n = X x 0 G h i I S i | finSupp 0 G h
14 4 5 6 7 13 eldprdi φ x S X G n I if n = X x 0 G G DProd S
15 12 14 eqeltrrd φ x S X x G DProd S
16 15 ex φ x S X x G DProd S
17 16 ssrdv φ S X G DProd S