Metamath Proof Explorer


Theorem ressinbas

Description: Restriction only cares about the part of the second set which intersects the base of the first. (Contributed by Stefan O'Rear, 29-Nov-2014)

Ref Expression
Hypothesis ressid.1 B = Base W
Assertion ressinbas A X W 𝑠 A = W 𝑠 A B

Proof

Step Hyp Ref Expression
1 ressid.1 B = Base W
2 elex A X A V
3 eqid W 𝑠 A = W 𝑠 A
4 3 1 ressid2 B A W V A V W 𝑠 A = W
5 ssid B B
6 incom A B = B A
7 df-ss B A B A = B
8 7 biimpi B A B A = B
9 6 8 syl5eq B A A B = B
10 5 9 sseqtrrid B A B A B
11 elex W V W V
12 inex1g A V A B V
13 eqid W 𝑠 A B = W 𝑠 A B
14 13 1 ressid2 B A B W V A B V W 𝑠 A B = W
15 10 11 12 14 syl3an B A W V A V W 𝑠 A B = W
16 4 15 eqtr4d B A W V A V W 𝑠 A = W 𝑠 A B
17 16 3expb B A W V A V W 𝑠 A = W 𝑠 A B
18 inass A B B = A B B
19 inidm B B = B
20 19 ineq2i A B B = A B
21 18 20 eqtr2i A B = A B B
22 21 opeq2i Base ndx A B = Base ndx A B B
23 22 oveq2i W sSet Base ndx A B = W sSet Base ndx A B B
24 3 1 ressval2 ¬ B A W V A V W 𝑠 A = W sSet Base ndx A B
25 inss1 A B A
26 sstr B A B A B A B A
27 25 26 mpan2 B A B B A
28 27 con3i ¬ B A ¬ B A B
29 13 1 ressval2 ¬ B A B W V A B V W 𝑠 A B = W sSet Base ndx A B B
30 28 11 12 29 syl3an ¬ B A W V A V W 𝑠 A B = W sSet Base ndx A B B
31 23 24 30 3eqtr4a ¬ B A W V A V W 𝑠 A = W 𝑠 A B
32 31 3expb ¬ B A W V A V W 𝑠 A = W 𝑠 A B
33 17 32 pm2.61ian W V A V W 𝑠 A = W 𝑠 A B
34 reldmress Rel dom 𝑠
35 34 ovprc1 ¬ W V W 𝑠 A =
36 34 ovprc1 ¬ W V W 𝑠 A B =
37 35 36 eqtr4d ¬ W V W 𝑠 A = W 𝑠 A B
38 37 adantr ¬ W V A V W 𝑠 A = W 𝑠 A B
39 33 38 pm2.61ian A V W 𝑠 A = W 𝑠 A B
40 2 39 syl A X W 𝑠 A = W 𝑠 A B