Theorem xporderlem 5996

Description: Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)

Hypothesis

Ref	Expression
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 ) ) ) ) }

Assertion

Ref	Expression
xporderlem	|- ( <. 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 ) ) ) )

Distinct variable groups:   x, A, y   x, B, y   x, R, y   x, S, y   x, a, y   x, b, y   x, c, y   x, d, y

Allowed substitution hints:   A( a, b, c, d)   B( a, b, c, d)   R( a, b, c, d)   S( a, b, c, d)   T( x, y, a, b, c, d)

Proof of Theorem xporderlem

Step	Hyp	Ref	Expression
1		df-br 3846	. . 3 |- ( <. a , b >. T <. c , d >. <-> <. <. a , b >. , <. c , d >. >. e. T )
2		xporderlem.1	. . . 4 |- 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 ) ) ) ) }
3	2	eleq2i 2154	. . 3 |- ( <. <. 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	1, 3	bitri 182	. 2 |- ( <. 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		vex 2622	. . . 4 |- a e. _V
6		vex 2622	. . . 4 |- b e. _V
7	5, 6	opex 4056	. . 3 |- <. a , b >. e. _V
8		vex 2622	. . . 4 |- c e. _V
9		vex 2622	. . . 4 |- d e. _V
10	8, 9	opex 4056	. . 3 |- <. c , d >. e. _V
11		eleq1 2150	. . . . . 6 |- ( x = <. a , b >. -> ( x e. ( A X. B ) <-> <. a , b >. e. ( A X. B ) ) )
12		opelxp 4467	. . . . . 6 |- ( <. a , b >. e. ( A X. B ) <-> ( a e. A /\ b e. B ) )
13	11, 12	syl6bb 194	. . . . 5 |- ( x = <. a , b >. -> ( x e. ( A X. B ) <-> ( a e. A /\ b e. B ) ) )
14	13	anbi1d 453	. . . 4 |- ( x = <. a , b >. -> ( ( x e. ( A X. B ) /\ y e. ( A X. B ) ) <-> ( ( a e. A /\ b e. B ) /\ y e. ( A X. B ) ) ) )
15	5, 6	op1std 5919	. . . . . 6 |- ( x = <. a , b >. -> ( 1st ` x ) = a )
16	15	breq1d 3855	. . . . 5 |- ( x = <. a , b >. -> ( ( 1st ` x ) R ( 1st ` y ) <-> a R ( 1st ` y ) ) )
17	15	eqeq1d 2096	. . . . . 6 |- ( x = <. a , b >. -> ( ( 1st ` x ) = ( 1st ` y ) <-> a = ( 1st ` y ) ) )
18	5, 6	op2ndd 5920	. . . . . . 7 |- ( x = <. a , b >. -> ( 2nd ` x ) = b )
19	18	breq1d 3855	. . . . . 6 |- ( x = <. a , b >. -> ( ( 2nd ` x ) S ( 2nd ` y ) <-> b S ( 2nd ` y ) ) )
20	17, 19	anbi12d 457	. . . . 5 |- ( x = <. a , b >. -> ( ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) <-> ( a = ( 1st ` y ) /\ b S ( 2nd ` y ) ) ) )
21	16, 20	orbi12d 742	. . . 4 |- ( 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 ) ) ) ) )
22	14, 21	anbi12d 457	. . 3 |- ( 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 ) ) ) ) ) )
23		eleq1 2150	. . . . . 6 |- ( y = <. c , d >. -> ( y e. ( A X. B ) <-> <. c , d >. e. ( A X. B ) ) )
24		opelxp 4467	. . . . . 6 |- ( <. c , d >. e. ( A X. B ) <-> ( c e. A /\ d e. B ) )
25	23, 24	syl6bb 194	. . . . 5 |- ( y = <. c , d >. -> ( y e. ( A X. B ) <-> ( c e. A /\ d e. B ) ) )
26	25	anbi2d 452	. . . 4 |- ( 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 ) ) ) )
27	8, 9	op1std 5919	. . . . . 6 |- ( y = <. c , d >. -> ( 1st ` y ) = c )
28	27	breq2d 3857	. . . . 5 |- ( y = <. c , d >. -> ( a R ( 1st ` y ) <-> a R c ) )
29	27	eqeq2d 2099	. . . . . 6 |- ( y = <. c , d >. -> ( a = ( 1st ` y ) <-> a = c ) )
30	8, 9	op2ndd 5920	. . . . . . 7 |- ( y = <. c , d >. -> ( 2nd ` y ) = d )
31	30	breq2d 3857	. . . . . 6 |- ( y = <. c , d >. -> ( b S ( 2nd ` y ) <-> b S d ) )
32	29, 31	anbi12d 457	. . . . 5 |- ( y = <. c , d >. -> ( ( a = ( 1st ` y ) /\ b S ( 2nd ` y ) ) <-> ( a = c /\ b S d ) ) )
33	28, 32	orbi12d 742	. . . 4 |- ( y = <. c , d >. -> ( ( a R ( 1st ` y ) \/ ( a = ( 1st ` y ) /\ b S ( 2nd ` y ) ) ) <-> ( a R c \/ ( a = c /\ b S d ) ) ) )
34	26, 33	anbi12d 457	. . 3 |- ( 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	7, 10, 22, 34	opelopab 4098	. 2 |- ( <. <. 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 553	. . 3 |- ( ( ( 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 446	. 2 |- ( ( ( ( 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 204	1 |- ( <. 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 ) ) ) )

Syntax hints:   /\ wa 102   <-> wb 103   \/ wo 664   = wceq 1289   e. wcel 1438   <.cop 3449   class class class wbr 3845   {copab 3898   X. cxp 4436   ` cfv 5015   1stc1st 5909   2ndc2nd 5910

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fv 5023  df-1st 5911  df-2nd 5912

This theorem is referenced by:  poxp  5997