Theorem xporderlem 7288

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 4654	. . 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 2693	. . 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 264	. 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		opex 4932	. . 3 |- <. a , b >. e. _V
6		opex 4932	. . 3 |- <. c , d >. e. _V
7		eleq1 2689	. . . . . 6 |- ( x = <. a , b >. -> ( x e. ( A X. B ) <-> <. a , b >. e. ( A X. B ) ) )
8		opelxp 5146	. . . . . 6 |- ( <. a , b >. e. ( A X. B ) <-> ( a e. A /\ b e. B ) )
9	7, 8	syl6bb 276	. . . . 5 |- ( x = <. a , b >. -> ( x e. ( A X. B ) <-> ( a e. A /\ b e. B ) ) )
10	9	anbi1d 741	. . . 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 ) ) ) )
11		vex 3203	. . . . . . 7 |- a e. _V
12		vex 3203	. . . . . . 7 |- b e. _V
13	11, 12	op1std 7178	. . . . . 6 |- ( x = <. a , b >. -> ( 1st ` x ) = a )
14	13	breq1d 4663	. . . . 5 |- ( x = <. a , b >. -> ( ( 1st ` x ) R ( 1st ` y ) <-> a R ( 1st ` y ) ) )
15	13	eqeq1d 2624	. . . . . 6 |- ( x = <. a , b >. -> ( ( 1st ` x ) = ( 1st ` y ) <-> a = ( 1st ` y ) ) )
16	11, 12	op2ndd 7179	. . . . . . 7 |- ( x = <. a , b >. -> ( 2nd ` x ) = b )
17	16	breq1d 4663	. . . . . 6 |- ( x = <. a , b >. -> ( ( 2nd ` x ) S ( 2nd ` y ) <-> b S ( 2nd ` y ) ) )
18	15, 17	anbi12d 747	. . . . 5 |- ( x = <. a , b >. -> ( ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) <-> ( a = ( 1st ` y ) /\ b S ( 2nd ` y ) ) ) )
19	14, 18	orbi12d 746	. . . 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 ) ) ) ) )
20	10, 19	anbi12d 747	. . 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 ) ) ) ) ) )
21		eleq1 2689	. . . . . 6 |- ( y = <. c , d >. -> ( y e. ( A X. B ) <-> <. c , d >. e. ( A X. B ) ) )
22		opelxp 5146	. . . . . 6 |- ( <. c , d >. e. ( A X. B ) <-> ( c e. A /\ d e. B ) )
23	21, 22	syl6bb 276	. . . . 5 |- ( y = <. c , d >. -> ( y e. ( A X. B ) <-> ( c e. A /\ d e. B ) ) )
24	23	anbi2d 740	. . . 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 ) ) ) )
25		vex 3203	. . . . . . 7 |- c e. _V
26		vex 3203	. . . . . . 7 |- d e. _V
27	25, 26	op1std 7178	. . . . . 6 |- ( y = <. c , d >. -> ( 1st ` y ) = c )
28	27	breq2d 4665	. . . . 5 |- ( y = <. c , d >. -> ( a R ( 1st ` y ) <-> a R c ) )
29	27	eqeq2d 2632	. . . . . 6 |- ( y = <. c , d >. -> ( a = ( 1st ` y ) <-> a = c ) )
30	25, 26	op2ndd 7179	. . . . . . 7 |- ( y = <. c , d >. -> ( 2nd ` y ) = d )
31	30	breq2d 4665	. . . . . 6 |- ( y = <. c , d >. -> ( b S ( 2nd ` y ) <-> b S d ) )
32	29, 31	anbi12d 747	. . . . 5 |- ( y = <. c , d >. -> ( ( a = ( 1st ` y ) /\ b S ( 2nd ` y ) ) <-> ( a = c /\ b S d ) ) )
33	28, 32	orbi12d 746	. . . 4 |- ( 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 747	. . 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	5, 6, 20, 34	opelopab 4997	. 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 865	. . 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 731	. 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 286	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:   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   {copab 4712   X. cxp 5112   ` cfv 5888   1stc1st 7166   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  poxp  7289  soxp  7290