Step |
Hyp |
Ref |
Expression |
1 |
|
xporderlem.1 |
|- T = { <. x , y >. | ( ( x e. ( A X. B ) /\ y e. ( A X. B ) ) /\ ( ( 1st ` x ) R ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) ) ) } |
2 |
|
df-br |
|- ( <. a , b >. T <. c , d >. <-> <. <. a , b >. , <. c , d >. >. e. T ) |
3 |
1
|
eleq2i |
|- ( <. <. a , b >. , <. c , d >. >. e. T <-> <. <. a , b >. , <. c , d >. >. e. { <. x , y >. | ( ( x e. ( A X. B ) /\ y e. ( A X. B ) ) /\ ( ( 1st ` x ) R ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) ) ) } ) |
4 |
2 3
|
bitri |
|- ( <. a , b >. T <. c , d >. <-> <. <. a , b >. , <. c , d >. >. e. { <. x , y >. | ( ( x e. ( A X. B ) /\ y e. ( A X. B ) ) /\ ( ( 1st ` x ) R ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) ) ) } ) |
5 |
|
opex |
|- <. a , b >. e. _V |
6 |
|
opex |
|- <. c , d >. e. _V |
7 |
|
eleq1 |
|- ( x = <. a , b >. -> ( x e. ( A X. B ) <-> <. a , b >. e. ( A X. B ) ) ) |
8 |
|
opelxp |
|- ( <. a , b >. e. ( A X. B ) <-> ( a e. A /\ b e. B ) ) |
9 |
7 8
|
bitrdi |
|- ( x = <. a , b >. -> ( x e. ( A X. B ) <-> ( a e. A /\ b e. B ) ) ) |
10 |
9
|
anbi1d |
|- ( x = <. a , b >. -> ( ( x e. ( A X. B ) /\ y e. ( A X. B ) ) <-> ( ( a e. A /\ b e. B ) /\ y e. ( A X. B ) ) ) ) |
11 |
|
vex |
|- a e. _V |
12 |
|
vex |
|- b e. _V |
13 |
11 12
|
op1std |
|- ( x = <. a , b >. -> ( 1st ` x ) = a ) |
14 |
13
|
breq1d |
|- ( x = <. a , b >. -> ( ( 1st ` x ) R ( 1st ` y ) <-> a R ( 1st ` y ) ) ) |
15 |
13
|
eqeq1d |
|- ( x = <. a , b >. -> ( ( 1st ` x ) = ( 1st ` y ) <-> a = ( 1st ` y ) ) ) |
16 |
11 12
|
op2ndd |
|- ( x = <. a , b >. -> ( 2nd ` x ) = b ) |
17 |
16
|
breq1d |
|- ( x = <. a , b >. -> ( ( 2nd ` x ) S ( 2nd ` y ) <-> b S ( 2nd ` y ) ) ) |
18 |
15 17
|
anbi12d |
|- ( x = <. a , b >. -> ( ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) <-> ( a = ( 1st ` y ) /\ b S ( 2nd ` y ) ) ) ) |
19 |
14 18
|
orbi12d |
|- ( x = <. a , b >. -> ( ( ( 1st ` x ) R ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) ) <-> ( a R ( 1st ` y ) \/ ( a = ( 1st ` y ) /\ b S ( 2nd ` y ) ) ) ) ) |
20 |
10 19
|
anbi12d |
|- ( x = <. a , b >. -> ( ( ( x e. ( A X. B ) /\ y e. ( A X. B ) ) /\ ( ( 1st ` x ) R ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) ) ) <-> ( ( ( a e. A /\ b e. B ) /\ y e. ( A X. B ) ) /\ ( a R ( 1st ` y ) \/ ( a = ( 1st ` y ) /\ b S ( 2nd ` y ) ) ) ) ) ) |
21 |
|
eleq1 |
|- ( y = <. c , d >. -> ( y e. ( A X. B ) <-> <. c , d >. e. ( A X. B ) ) ) |
22 |
|
opelxp |
|- ( <. c , d >. e. ( A X. B ) <-> ( c e. A /\ d e. B ) ) |
23 |
21 22
|
bitrdi |
|- ( y = <. c , d >. -> ( y e. ( A X. B ) <-> ( c e. A /\ d e. B ) ) ) |
24 |
23
|
anbi2d |
|- ( y = <. c , d >. -> ( ( ( a e. A /\ b e. B ) /\ y e. ( A X. B ) ) <-> ( ( a e. A /\ b e. B ) /\ ( c e. A /\ d e. B ) ) ) ) |
25 |
|
vex |
|- c e. _V |
26 |
|
vex |
|- d e. _V |
27 |
25 26
|
op1std |
|- ( y = <. c , d >. -> ( 1st ` y ) = c ) |
28 |
27
|
breq2d |
|- ( y = <. c , d >. -> ( a R ( 1st ` y ) <-> a R c ) ) |
29 |
27
|
eqeq2d |
|- ( y = <. c , d >. -> ( a = ( 1st ` y ) <-> a = c ) ) |
30 |
25 26
|
op2ndd |
|- ( y = <. c , d >. -> ( 2nd ` y ) = d ) |
31 |
30
|
breq2d |
|- ( y = <. c , d >. -> ( b S ( 2nd ` y ) <-> b S d ) ) |
32 |
29 31
|
anbi12d |
|- ( y = <. c , d >. -> ( ( a = ( 1st ` y ) /\ b S ( 2nd ` y ) ) <-> ( a = c /\ b S d ) ) ) |
33 |
28 32
|
orbi12d |
|- ( y = <. c , d >. -> ( ( a R ( 1st ` y ) \/ ( a = ( 1st ` y ) /\ b S ( 2nd ` y ) ) ) <-> ( a R c \/ ( a = c /\ b S d ) ) ) ) |
34 |
24 33
|
anbi12d |
|- ( y = <. c , d >. -> ( ( ( ( a e. A /\ b e. B ) /\ y e. ( A X. B ) ) /\ ( a R ( 1st ` y ) \/ ( a = ( 1st ` y ) /\ b S ( 2nd ` y ) ) ) ) <-> ( ( ( a e. A /\ b e. B ) /\ ( c e. A /\ d e. B ) ) /\ ( a R c \/ ( a = c /\ b S d ) ) ) ) ) |
35 |
5 6 20 34
|
opelopab |
|- ( <. <. a , b >. , <. c , d >. >. e. { <. x , y >. | ( ( x e. ( A X. B ) /\ y e. ( A X. B ) ) /\ ( ( 1st ` x ) R ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) ) ) } <-> ( ( ( a e. A /\ b e. B ) /\ ( c e. A /\ d e. B ) ) /\ ( a R c \/ ( a = c /\ b S d ) ) ) ) |
36 |
|
an4 |
|- ( ( ( a e. A /\ b e. B ) /\ ( c e. A /\ d e. B ) ) <-> ( ( a e. A /\ c e. A ) /\ ( b e. B /\ d e. B ) ) ) |
37 |
36
|
anbi1i |
|- ( ( ( ( a e. A /\ b e. B ) /\ ( c e. A /\ d e. B ) ) /\ ( a R c \/ ( a = c /\ b S d ) ) ) <-> ( ( ( a e. A /\ c e. A ) /\ ( b e. B /\ d e. B ) ) /\ ( a R c \/ ( a = c /\ b S d ) ) ) ) |
38 |
4 35 37
|
3bitri |
|- ( <. a , b >. T <. c , d >. <-> ( ( ( a e. A /\ c e. A ) /\ ( b e. B /\ d e. B ) ) /\ ( a R c \/ ( a = c /\ b S d ) ) ) ) |