Metamath Proof Explorer


Theorem ex-opab

Description: Example for df-opab . Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015)

Ref Expression
Assertion ex-opab R=xy|xyx+1=y3R4

Proof

Step Hyp Ref Expression
1 3cn 3
2 4cn 4
3 3p1e4 3+1=4
4 1 elexi 3V
5 2 elexi 4V
6 eleq1 x=3x3
7 oveq1 x=3x+1=3+1
8 7 eqeq1d x=3x+1=y3+1=y
9 6 8 3anbi13d x=3xyx+1=y3y3+1=y
10 eleq1 y=4y4
11 eqeq2 y=43+1=y3+1=4
12 10 11 3anbi23d y=43y3+1=y343+1=4
13 eqid xy|xyx+1=y=xy|xyx+1=y
14 4 5 9 12 13 brab 3xy|xyx+1=y4343+1=4
15 1 2 3 14 mpbir3an 3xy|xyx+1=y4
16 breq R=xy|xyx+1=y3R43xy|xyx+1=y4
17 15 16 mpbiri R=xy|xyx+1=y3R4