Theorem xporderlem 6395

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 4089	. . 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 2298	. . 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 184	. 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 2805	. . . 4 |- a e. _V
6		vex 2805	. . . 4 |- b e. _V
7	5, 6	opex 4321	. . 3 |- <. a , b >. e. _V
8		vex 2805	. . . 4 |- c e. _V
9		vex 2805	. . . 4 |- d e. _V
10	8, 9	opex 4321	. . 3 |- <. c , d >. e. _V
11		eleq1 2294	. . . . . 6 |- ( x = <. a , b >. -> ( x e. ( A X. B ) <-> <. a , b >. e. ( A X. B ) ) )
12		opelxp 4755	. . . . . 6 |- ( <. a , b >. e. ( A X. B ) <-> ( a e. A /\ b e. B ) )
13	11, 12	bitrdi 196	. . . . 5 |- ( x = <. a , b >. -> ( x e. ( A X. B ) <-> ( a e. A /\ b e. B ) ) )
14	13	anbi1d 465	. . . 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 6310	. . . . . 6 |- ( x = <. a , b >. -> ( 1st ` x ) = a )
16	15	breq1d 4098	. . . . 5 |- ( x = <. a , b >. -> ( ( 1st ` x ) R ( 1st ` y ) <-> a R ( 1st ` y ) ) )
17	15	eqeq1d 2240	. . . . . 6 |- ( x = <. a , b >. -> ( ( 1st ` x ) = ( 1st ` y ) <-> a = ( 1st ` y ) ) )
18	5, 6	op2ndd 6311	. . . . . . 7 |- ( x = <. a , b >. -> ( 2nd ` x ) = b )
19	18	breq1d 4098	. . . . . 6 |- ( x = <. a , b >. -> ( ( 2nd ` x ) S ( 2nd ` y ) <-> b S ( 2nd ` y ) ) )
20	17, 19	anbi12d 473	. . . . 5 |- ( x = <. a , b >. -> ( ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) <-> ( a = ( 1st ` y ) /\ b S ( 2nd ` y ) ) ) )
21	16, 20	orbi12d 800	. . . 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 473	. . 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 2294	. . . . . 6 |- ( y = <. c , d >. -> ( y e. ( A X. B ) <-> <. c , d >. e. ( A X. B ) ) )
24		opelxp 4755	. . . . . 6 |- ( <. c , d >. e. ( A X. B ) <-> ( c e. A /\ d e. B ) )
25	23, 24	bitrdi 196	. . . . 5 |- ( y = <. c , d >. -> ( y e. ( A X. B ) <-> ( c e. A /\ d e. B ) ) )
26	25	anbi2d 464	. . . 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 6310	. . . . . 6 |- ( y = <. c , d >. -> ( 1st ` y ) = c )
28	27	breq2d 4100	. . . . 5 |- ( y = <. c , d >. -> ( a R ( 1st ` y ) <-> a R c ) )
29	27	eqeq2d 2243	. . . . . 6 |- ( y = <. c , d >. -> ( a = ( 1st ` y ) <-> a = c ) )
30	8, 9	op2ndd 6311	. . . . . . 7 |- ( y = <. c , d >. -> ( 2nd ` y ) = d )
31	30	breq2d 4100	. . . . . 6 |- ( y = <. c , d >. -> ( b S ( 2nd ` y ) <-> b S d ) )
32	29, 31	anbi12d 473	. . . . 5 |- ( y = <. c , d >. -> ( ( a = ( 1st ` y ) /\ b S ( 2nd ` y ) ) <-> ( a = c /\ b S d ) ) )
33	28, 32	orbi12d 800	. . . 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 473	. . 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 4366	. 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 588	. . 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 458	. 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 206	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 104   <-> wb 105   \/ wo 715   = wceq 1397   e. wcel 2202   <.cop 3672   class class class wbr 4088   {copab 4149   X. cxp 4723   ` cfv 5326   1stc1st 6300   2ndc2nd 6301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fv 5334  df-1st 6302  df-2nd 6303

This theorem is referenced by:  poxp  6396