Theorem unixp 3517

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 2179	. . . 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 3332	. . . 4 |- (B =/= (/) -> dom ( A X. B) = A)
3		rnxp 3472	. . . 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 3201	. . . . 5 |- (B = (/) -> (A X. B) = (A X. (/)))
6		xp0 3465	. . . . 5 |- (A X. (/)) = (/)
7	5, 6	syl6eq 1523	. . . 4 |- (B = (/) -> (A X. B) = (/))
8	7	necon3i 1605	. . 3 |- ((A X. B) =/= (/) -> B =/= (/))
9		xpeq1 3200	. . . . 5 |- (A = (/) -> (A X. B) = ((/) X. B))
10		xp0r 3239	. . . . 5 |- ((/) X. B) = (/)
11	9, 10	syl6eq 1523	. . . 4 |- (A = (/) -> (A X. B) = (/))
12	11	necon3i 1605	. . 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 3255	. . 3 |- Rel (A X. B)
15		relfld 3515	. . 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 1519	1 |- ((A X. B) =/= (/) -> U.U.(A X. B) = (A u. B))

Syntax hints:   -> wi 3   = wceq 956   =/= wne 1585   u. cun 2045  (/)c0 2280  U.cuni 2503   X. cxp 3168  dom cdm 3170  ran crn 3171  Rel wrel 3175

This theorem is referenced by:  rankxpl 4710  rankxplim2 4713  rankxplim3 4714  rankxpsuc 4715

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189

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