Theorem unixp 4422

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		xpeq2 4017	. . . . 5 |- (B = (/) -> (A X. B) = (A X. (/)))
2		xp0 4334	. . . . 5 |- (A X. (/)) = (/)
3	1, 2	syl6eq 1944	. . . 4 |- (B = (/) -> (A X. B) = (/))
4	3	necon3i 2042	. . 3 |- ((A X. B) =/= (/) -> B =/= (/))
5		xpeq1 4016	. . . . 5 |- (A = (/) -> (A X. B) = ((/) X. B))
6		xp0r 4065	. . . . 5 |- ((/) X. B) = (/)
7	5, 6	syl6eq 1944	. . . 4 |- (A = (/) -> (A X. B) = (/))
8	7	necon3i 2042	. . 3 |- ((A X. B) =/= (/) -> A =/= (/))
9		uneq12 2750	. . . 4 |- ((dom ( A X. B) = A /\ ran ( A X. B) = B) -> (dom ( A X. B) u. ran ( A X. B)) = (A u. B))
10		dmxp 4177	. . . 4 |- (B =/= (/) -> dom ( A X. B) = A)
11		rnxp 4342	. . . 4 |- (A =/= (/) -> ran ( A X. B) = B)
12	9, 10, 11	syl2an 503	. . 3 |- ((B =/= (/) /\ A =/= (/)) -> (dom ( A X. B) u. ran ( A X. B)) = (A u. B))
13	4, 8, 12	syl11anc 524	. 2 |- ((A X. B) =/= (/) -> (dom ( A X. B) u. ran ( A X. B)) = (A u. B))
14		relxp 4088	. . 3 |- Rel (A X. B)
15		relfld 4419	. . 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 1940	1 |- ((A X. B) =/= (/) -> U.U.(A X. B) = (A u. B))

Syntax hints:   -> wi 3   = wceq 1298   =/= wne 2017   u. cun 2591  (/)c0 2875  U.cuni 3177   X. cxp 3984  dom cdm 3986  ran crn 3987  Rel wrel 3991

This theorem is referenced by:  rankxpl 5821  rankxplim2 5824  rankxplim3 5825  rankxpsuc 5826  fldsqcp2 14378

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-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

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-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005

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