Theorem unixp 3503

Description: The double class union of a non-empty cross product is the union of it members.

Assertion

Ref	Expression
unixp	|- ((A X. B) =/= (/) -> U.U.(A X. B) = (A u. B))

Proof of Theorem unixp

Step	Hyp	Ref	Expression
1		uneq12 2169	. . . 4 |- ((dom ( A X. B) = A /\ ran ( A X. B) = B) -> (dom ( A X. B) u. ran ( A X. B)) = (A u. B))
2		dmxp 3321	. . . 4 |- (B =/= (/) -> dom ( A X. B) = A)
3		rnxp 3458	. . . 4 |- (A =/= (/) -> ran ( A X. B) = B)
4	1, 2, 3	syl2an 454	. . 3 |- ((B =/= (/) /\ A =/= (/)) -> (dom ( A X. B) u. ran ( A X. B)) = (A u. B))
5		xpeq2 3191	. . . . 5 |- (B = (/) -> (A X. B) = (A X. (/)))
6		xp0 3451	. . . . 5 |- (A X. (/)) = (/)
7	5, 6	syl6eq 1515	. . . 4 |- (B = (/) -> (A X. B) = (/))
8	7	necon3i 1597	. . 3 |- ((A X. B) =/= (/) -> B =/= (/))
9		xpeq1 3190	. . . . 5 |- (A = (/) -> (A X. B) = ((/) X. B))
10		xp0r 3229	. . . . 5 |- ((/) X. B) = (/)
11	9, 10	syl6eq 1515	. . . 4 |- (A = (/) -> (A X. B) = (/))
12	11	necon3i 1597	. . 3 |- ((A X. B) =/= (/) -> A =/= (/))
13	4, 8, 12	sylanc 471	. 2 |- ((A X. B) =/= (/) -> (dom ( A X. B) u. ran ( A X. B)) = (A u. B))
14		relxp 3245	. . 3 |- Rel (A X. B)
15		relfld 3501	. . 3 |- (Rel (A X. B) -> U.U.(A X. B) = (dom ( A X. B) u. ran ( A X. B)))
16	14, 15	ax-mp 7	. 2 |- U.U.(A X. B) = (dom ( A X. B) u. ran ( A X. B))
17	13, 16	syl5eq 1511	1 |- ((A X. B) =/= (/) -> U.U.(A X. B) = (A u. B))

Syntax hints:   -> wi 3   = wceq 953   =/= wne 1577   u. cun 2035  (/)c0 2270  U.cuni 2493   X. cxp 3158  dom cdm 3160  ran crn 3161  Rel wrel 3165

This theorem is referenced by:  rankxpl 4682  rankxplim2 4685  rankxplim3 4686  rankxpsuc 4687

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-pow 2732  ax-pr 2769

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-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-xp 3174  df-rel 3175  df-cnv 3176  df-dm 3178  df-rn 3179

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