Theorem unielxp 5047

Description: The membership relation for a cross product is inherited by union.

Assertion

Ref	Expression
unielxp	|- (A e. (B X. C) -> U.A e. U.(B X. C))

Proof of Theorem unielxp

Step	Hyp	Ref	Expression
1		elxp7 5042	. 2 |- (A e. (B X. C) <-> (A e. (_V X. _V) /\ ((1st` A) e. B /\ (2nd` A) e. C)))
2		elvvuni 4055	. . . 4 |- (A e. (_V X. _V) -> U.A e. A)
3	2	adantr 425	. . 3 |- ((A e. (_V X. _V) /\ ((1st` A) e. B /\ (2nd` A) e. C)) -> U.A e. A)
4		simprl 450	. . . . . 6 |- ((U.A e. A /\ (A e. (_V X. _V) /\ ((1st` A) e. B /\ (2nd` A) e. C))) -> A e. (_V X. _V))
5		eleq2 1958	. . . . . . . 8 |- (x = A -> (U.A e. x <-> U.A e. A))
6		eleq1 1957	. . . . . . . . 9 |- (x = A -> (x e. (_V X. _V) <-> A e. (_V X. _V)))
7		fveq2 4681	. . . . . . . . . . 11 |- (x = A -> (1st` x) = (1st` A))
8	7	eleq1d 1963	. . . . . . . . . 10 |- (x = A -> ((1st` x) e. B <-> (1st` A) e. B))
9		fveq2 4681	. . . . . . . . . . 11 |- (x = A -> (2nd` x) = (2nd` A))
10	9	eleq1d 1963	. . . . . . . . . 10 |- (x = A -> ((2nd` x) e. C <-> (2nd` A) e. C))
11	8, 10	anbi12d 690	. . . . . . . . 9 |- (x = A -> (((1st` x) e. B /\ (2nd` x) e. C) <-> ((1st` A) e. B /\ (2nd` A) e. C)))
12	6, 11	anbi12d 690	. . . . . . . 8 |- (x = A -> ((x e. (_V X. _V) /\ ((1st` x) e. B /\ (2nd` x) e. C)) <-> (A e. (_V X. _V) /\ ((1st` A) e. B /\ (2nd` A) e. C))))
13	5, 12	anbi12d 690	. . . . . . 7 |- (x = A -> ((U.A e. x /\ (x e. (_V X. _V) /\ ((1st` x) e. B /\ (2nd` x) e. C))) <-> (U.A e. A /\ (A e. (_V X. _V) /\ ((1st` A) e. B /\ (2nd` A) e. C)))))
14	13	cla4egv 2365	. . . . . 6 |- (A e. (_V X. _V) -> ((U.A e. A /\ (A e. (_V X. _V) /\ ((1st` A) e. B /\ (2nd` A) e. C))) -> E.x(U.A e. x /\ (x e. (_V X. _V) /\ ((1st` x) e. B /\ (2nd` x) e. C)))))
15	4, 14	mpcom 60	. . . . 5 |- ((U.A e. A /\ (A e. (_V X. _V) /\ ((1st` A) e. B /\ (2nd` A) e. C))) -> E.x(U.A e. x /\ (x e. (_V X. _V) /\ ((1st` x) e. B /\ (2nd` x) e. C))))
16		eluniab 3189	. . . . 5 |- (U.A e. U.{x | (x e. (_V X. _V) /\ ((1st` x) e. B /\ (2nd` x) e. C))} <-> E.x(U.A e. x /\ (x e. (_V X. _V) /\ ((1st` x) e. B /\ (2nd` x) e. C))))
17	15, 16	sylibr 217	. . . 4 |- ((U.A e. A /\ (A e. (_V X. _V) /\ ((1st` A) e. B /\ (2nd` A) e. C))) -> U.A e. U.{x | (x e. (_V X. _V) /\ ((1st` x) e. B /\ (2nd` x) e. C))})
18		xp2 5045	. . . . . 6 |- (B X. C) = {x e. (_V X. _V) | ((1st` x) e. B /\ (2nd` x) e. C)}
19		df-rab 2112	. . . . . 6 |- {x e. (_V X. _V) | ((1st` x) e. B /\ (2nd` x) e. C)} = {x | (x e. (_V X. _V) /\ ((1st` x) e. B /\ (2nd` x) e. C))}
20	18, 19	eqtri 1908	. . . . 5 |- (B X. C) = {x | (x e. (_V X. _V) /\ ((1st` x) e. B /\ (2nd` x) e. C))}
21	20	unieqi 3187	. . . 4 |- U.(B X. C) = U.{x | (x e. (_V X. _V) /\ ((1st` x) e. B /\ (2nd` x) e. C))}
22	17, 21	syl6eleqr 1982	. . 3 |- ((U.A e. A /\ (A e. (_V X. _V) /\ ((1st` A) e. B /\ (2nd` A) e. C))) -> U.A e. U.(B X. C))
23	3, 22	mpancom 769	. 2 |- ((A e. (_V X. _V) /\ ((1st` A) e. B /\ (2nd` A) e. C)) -> U.A e. U.(B X. C))
24	1, 23	sylbi 216	1 |- (A e. (B X. C) -> U.A e. U.(B X. C))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  {crab 2108  _Vcvv 2292  U.cuni 3177   X. cxp 3984  ` cfv 3998  1stc1st 5018  2ndc2nd 5019

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-rab 2112  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

Copyright terms: Public domain