Theorem unielxp 4091

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 4087	. 2 |- (A e. (B X. C) <-> (A e. (V X. V) /\ ((1st` A) e. B /\ (2nd` A) e. C)))
2		elvvuni 3219	. . . 4 |- (A e. (V X. V) -> U.A e. A)
3	2	adantr 389	. . 3 |- ((A e. (V X. V) /\ ((1st` A) e. B /\ (2nd` A) e. C)) -> U.A e. A)
4		simprl 414	. . . . . 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 1527	. . . . . . . 8 |- (x = A -> (U.A e. x <-> U.A e. A))
6		eleq1 1526	. . . . . . . . 9 |- (x = A -> (x e. (V X. V) <-> A e. (V X. V)))
7		fveq2 3709	. . . . . . . . . . 11 |- (x = A -> (1st` x) = (1st` A))
8	7	eleq1d 1532	. . . . . . . . . 10 |- (x = A -> ((1st` x) e. B <-> (1st` A) e. B))
9		fveq2 3709	. . . . . . . . . . 11 |- (x = A -> (2nd` x) = (2nd` A))
10	9	eleq1d 1532	. . . . . . . . . 10 |- (x = A -> ((2nd` x) e. C <-> (2nd` A) e. C))
11	8, 10	anbi12d 626	. . . . . . . . 9 |- (x = A -> (((1st` x) e. B /\ (2nd` x) e. C) <-> ((1st` A) e. B /\ (2nd` A) e. C)))
12	6, 11	anbi12d 626	. . . . . . . 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 626	. . . . . . 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 1854	. . . . . 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 49	. . . . 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 2503	. . . . 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 200	. . . 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 4089	. . . . . 6 |- (B X. C) = {x e. (V X. V) | ((1st` x) e. B /\ (2nd` x) e. C)}
19		df-rab 1644	. . . . . 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	eqtr 1487	. . . . 5 |- (B X. C) = {x | (x e. (V X. V) /\ ((1st` x) e. B /\ (2nd` x) e. C))}
21	20	unieqi 2501	. . . 4 |- U.(B X. C) = U.{x | (x e. (V X. V) /\ ((1st` x) e. B /\ (2nd` x) e. C))}
22	17, 21	syl6eleqr 1551	. . 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 703	. 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 199	1 |- (A e. (B X. C) -> U.A e. U.(B X. C))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  {cab 1456  {crab 1640  Vcvv 1802  U.cuni 2493   X. cxp 3158  ` cfv 3172  1stc1st 4061  2ndc2nd 4062

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fv 3188  df-1st 4063  df-2nd 4064

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