Theorem xporderlem 13948

Description: Lemma for the lexicographical ordering theorems.

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)) /\ (aRc \/ (a = c /\ bSd))))

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

Proof of Theorem xporderlem

Step	Hyp	Ref	Expression
1		df-br 3339	. . 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 1961	. . 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 190	. 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 3527	. . 3 |- <.a, b>. e. _V
6		opex 3527	. . 3 |- <.c, d>. e. _V
7		eleq1 1957	. . . . . 6 |- (x = <.a, b>. -> (x e. (A X. B) <-> <.a, b>. e. (A X. B)))
8		visset 2295	. . . . . . 7 |- b e. _V
9	8	opelxp 4036	. . . . . 6 |- (<.a, b>. e. (A X. B) <-> (a e. A /\ b e. B))
10	7, 9	syl6bb 595	. . . . 5 |- (x = <.a, b>. -> (x e. (A X. B) <-> (a e. A /\ b e. B)))
11	10	anbi1d 679	. . . 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))))
12		fveq2 4681	. . . . . . 7 |- (x = <.a, b>. -> (1st` x) = (1st` <.a, b>.))
13		visset 2295	. . . . . . . 8 |- a e. _V
14	13	op1st 5026	. . . . . . 7 |- (1st` <.a, b>.) = a
15	12, 14	syl6eq 1944	. . . . . 6 |- (x = <.a, b>. -> (1st` x) = a)
16	15	breq1d 3348	. . . . 5 |- (x = <.a, b>. -> ((1st` x)R(1st` y) <-> aR(1st` y)))
17	15	eqeq1d 1892	. . . . . 6 |- (x = <.a, b>. -> ((1st` x) = (1st` y) <-> a = (1st` y)))
18		fveq2 4681	. . . . . . . 8 |- (x = <.a, b>. -> (2nd` x) = (2nd` <.a, b>.))
19	13, 8	op2nd 5027	. . . . . . . 8 |- (2nd` <.a, b>.) = b
20	18, 19	syl6eq 1944	. . . . . . 7 |- (x = <.a, b>. -> (2nd` x) = b)
21	20	breq1d 3348	. . . . . 6 |- (x = <.a, b>. -> ((2nd` x)S(2nd` y) <-> bS(2nd` y)))
22	17, 21	anbi12d 690	. . . . 5 |- (x = <.a, b>. -> (((1st` x) = (1st` y) /\ (2nd` x)S(2nd` y)) <-> (a = (1st` y) /\ bS(2nd` y))))
23	16, 22	orbi12d 689	. . . 4 |- (x = <.a, b>. -> (((1st` x)R(1st` y) \/ ((1st` x) = (1st` y) /\ (2nd` x)S(2nd` y))) <-> (aR(1st` y) \/ (a = (1st` y) /\ bS(2nd` y)))))
24	11, 23	anbi12d 690	. . 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)) /\ (aR(1st` y) \/ (a = (1st` y) /\ bS(2nd` y))))))
25		eleq1 1957	. . . . . 6 |- (y = <.c, d>. -> (y e. (A X. B) <-> <.c, d>. e. (A X. B)))
26		visset 2295	. . . . . . 7 |- d e. _V
27	26	opelxp 4036	. . . . . 6 |- (<.c, d>. e. (A X. B) <-> (c e. A /\ d e. B))
28	25, 27	syl6bb 595	. . . . 5 |- (y = <.c, d>. -> (y e. (A X. B) <-> (c e. A /\ d e. B)))
29	28	anbi2d 678	. . . 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))))
30		fveq2 4681	. . . . . . 7 |- (y = <.c, d>. -> (1st` y) = (1st` <.c, d>.))
31		visset 2295	. . . . . . . 8 |- c e. _V
32	31	op1st 5026	. . . . . . 7 |- (1st` <.c, d>.) = c
33	30, 32	syl6eq 1944	. . . . . 6 |- (y = <.c, d>. -> (1st` y) = c)
34	33	breq2d 3350	. . . . 5 |- (y = <.c, d>. -> (aR(1st` y) <-> aRc))
35	33	eqeq2d 1895	. . . . . 6 |- (y = <.c, d>. -> (a = (1st` y) <-> a = c))
36		fveq2 4681	. . . . . . . 8 |- (y = <.c, d>. -> (2nd` y) = (2nd` <.c, d>.))
37	31, 26	op2nd 5027	. . . . . . . 8 |- (2nd` <.c, d>.) = d
38	36, 37	syl6eq 1944	. . . . . . 7 |- (y = <.c, d>. -> (2nd` y) = d)
39	38	breq2d 3350	. . . . . 6 |- (y = <.c, d>. -> (bS(2nd` y) <-> bSd))
40	35, 39	anbi12d 690	. . . . 5 |- (y = <.c, d>. -> ((a = (1st` y) /\ bS(2nd` y)) <-> (a = c /\ bSd)))
41	34, 40	orbi12d 689	. . . 4 |- (y = <.c, d>. -> ((aR(1st` y) \/ (a = (1st` y) /\ bS(2nd` y))) <-> (aRc \/ (a = c /\ bSd))))
42	29, 41	anbi12d 690	. . 3 |- (y = <.c, d>. -> ((((a e. A /\ b e. B) /\ y e. (A X. B)) /\ (aR(1st` y) \/ (a = (1st` y) /\ bS(2nd` y)))) <-> (((a e. A /\ b e. B) /\ (c e. A /\ d e. B)) /\ (aRc \/ (a = c /\ bSd)))))
43	5, 6, 24, 42	opelopab 3570	. 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)) /\ (aRc \/ (a = c /\ bSd))))
44		an4 564	. . 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)))
45	44	anbi1i 539	. 2 |- ((((a e. A /\ b e. B) /\ (c e. A /\ d e. B)) /\ (aRc \/ (a = c /\ bSd))) <-> (((a e. A /\ c e. A) /\ (b e. B /\ d e. B)) /\ (aRc \/ (a = c /\ bSd))))
46	4, 43, 45	3bitri 194	1 |- (<.a, b>.T<.c, d>. <-> (((a e. A /\ c e. A) /\ (b e. B /\ d e. B)) /\ (aRc \/ (a = c /\ bSd))))

Syntax hints:   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  <.cop 3046   class class class wbr 3338  {copab 3395   X. cxp 3984  ` cfv 3998  1stc1st 5018  2ndc2nd 5019

This theorem is referenced by:  poxp 13949  soxp 13950

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-1st 5020  df-2nd 5021

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