Metamath Proof Explorer


Theorem brcap

Description: Binary relation form of the Cap function. (Contributed by Scott Fenton, 17-Apr-2014) (Revised by Mario Carneiro, 19-Apr-2014)

Ref Expression
Hypotheses brcap.1 A V
brcap.2 B V
brcap.3 C V
Assertion brcap A B 𝖢𝖺𝗉 C C = A B

Proof

Step Hyp Ref Expression
1 brcap.1 A V
2 brcap.2 B V
3 brcap.3 C V
4 opex A B V
5 df-cap 𝖢𝖺𝗉 = V × V × V ran V E 1 st -1 E 2 nd -1 E V
6 1 2 opelvv A B V × V
7 brxp A B V × V × V C A B V × V C V
8 6 3 7 mpbir2an A B V × V × V C
9 epel x E y x y
10 vex y V
11 10 4 brcnv y 1 st -1 A B A B 1 st y
12 1 2 br1steq A B 1 st y y = A
13 11 12 bitri y 1 st -1 A B y = A
14 9 13 anbi12ci x E y y 1 st -1 A B y = A x y
15 14 exbii y x E y y 1 st -1 A B y y = A x y
16 vex x V
17 16 4 brco x 1 st -1 E A B y x E y y 1 st -1 A B
18 1 clel3 x A y y = A x y
19 15 17 18 3bitr4i x 1 st -1 E A B x A
20 10 4 brcnv y 2 nd -1 A B A B 2 nd y
21 1 2 br2ndeq A B 2 nd y y = B
22 20 21 bitri y 2 nd -1 A B y = B
23 9 22 anbi12ci x E y y 2 nd -1 A B y = B x y
24 23 exbii y x E y y 2 nd -1 A B y y = B x y
25 16 4 brco x 2 nd -1 E A B y x E y y 2 nd -1 A B
26 2 clel3 x B y y = B x y
27 24 25 26 3bitr4i x 2 nd -1 E A B x B
28 19 27 anbi12i x 1 st -1 E A B x 2 nd -1 E A B x A x B
29 brin x 1 st -1 E 2 nd -1 E A B x 1 st -1 E A B x 2 nd -1 E A B
30 elin x A B x A x B
31 28 29 30 3bitr4ri x A B x 1 st -1 E 2 nd -1 E A B
32 4 3 5 8 31 brtxpsd3 A B 𝖢𝖺𝗉 C C = A B