Theorem xporderlem 6226

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 4001	. . 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 2244	. . 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 2740	. . . 4 |- a e. _V
6		vex 2740	. . . 4 |- b e. _V
7	5, 6	opex 4226	. . 3 |- <. a , b >. e. _V
8		vex 2740	. . . 4 |- c e. _V
9		vex 2740	. . . 4 |- d e. _V
10	8, 9	opex 4226	. . 3 |- <. c , d >. e. _V
11		eleq1 2240	. . . . . 6 |- ( x = <. a , b >. -> ( x e. ( A X. B ) <-> <. a , b >. e. ( A X. B ) ) )
12		opelxp 4653	. . . . . 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 6143	. . . . . 6 |- ( x = <. a , b >. -> ( 1st ` x ) = a )
16	15	breq1d 4010	. . . . 5 |- ( x = <. a , b >. -> ( ( 1st ` x ) R ( 1st ` y ) <-> a R ( 1st ` y ) ) )
17	15	eqeq1d 2186	. . . . . 6 |- ( x = <. a , b >. -> ( ( 1st ` x ) = ( 1st ` y ) <-> a = ( 1st ` y ) ) )
18	5, 6	op2ndd 6144	. . . . . . 7 |- ( x = <. a , b >. -> ( 2nd ` x ) = b )
19	18	breq1d 4010	. . . . . 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 793	. . . 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 2240	. . . . . 6 |- ( y = <. c , d >. -> ( y e. ( A X. B ) <-> <. c , d >. e. ( A X. B ) ) )
24		opelxp 4653	. . . . . 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 6143	. . . . . 6 |- ( y = <. c , d >. -> ( 1st ` y ) = c )
28	27	breq2d 4012	. . . . 5 |- ( y = <. c , d >. -> ( a R ( 1st ` y ) <-> a R c ) )
29	27	eqeq2d 2189	. . . . . 6 |- ( y = <. c , d >. -> ( a = ( 1st ` y ) <-> a = c ) )
30	8, 9	op2ndd 6144	. . . . . . 7 |- ( y = <. c , d >. -> ( 2nd ` y ) = d )
31	30	breq2d 4012	. . . . . 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 793	. . . 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 4268	. 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 586	. . 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 708   = wceq 1353   e. wcel 2148   <.cop 3594   class class class wbr 4000   {copab 4060   X. cxp 4621   ` cfv 5212   1stc1st 6133   2ndc2nd 6134

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-iota 5174  df-fun 5214  df-fv 5220  df-1st 6135  df-2nd 6136

This theorem is referenced by:  poxp  6227