Theorem dmxp 4177

Description: The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
dmxp	|- (B =/= (/) -> dom ( A X. B) = A)

Proof of Theorem dmxp

Step	Hyp	Ref	Expression
1		n0 2884	. . . . . 6 |- (B =/= (/) <-> E.x x e. B)
2	1	biimpi 168	. . . . 5 |- (B =/= (/) -> E.x x e. B)
3	2	a1d 15	. . . 4 |- (B =/= (/) -> (y e. A -> E.x x e. B))
4	3	r19.21aiv 2175	. . 3 |- (B =/= (/) -> A.y e. A E.x x e. B)
5		dmopab3 4169	. . 3 |- (A.y e. A E.x x e. B <-> dom {<.y, x>. | (y e. A /\ x e. B)} = A)
6	4, 5	sylib 215	. 2 |- (B =/= (/) -> dom {<.y, x>. | (y e. A /\ x e. B)} = A)
7		df-xp 4000	. . 3 |- (A X. B) = {<.y, x>. | (y e. A /\ x e. B)}
8	7	dmeqi 4158	. 2 |- dom ( A X. B) = dom {<.y, x>. | (y e. A /\ x e. B)}
9	6, 8	syl5eq 1940	1 |- (B =/= (/) -> dom ( A X. B) = A)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105  (/)c0 2875  {copab 3395   X. cxp 3984  dom cdm 3986

This theorem is referenced by:  dmxpid 4179  rnxp 4342  dmxpss 4343  ssxpb 4346  xpexr2 4353  relrelss 4417  unixp 4422  fconstOLD 4603  fodomr 5547  climuz0i 8368  ismgm 10367  frxp 13951  axbday 14012  axfelem2 14032  prjcp1 14399  cur1vald 14547  valcurfn1 14552  rngmgmbs3 14766  heiborlem18 15972  bfp 16009  rrndm 16015  xpexcnv 16436

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-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-br 3339  df-opab 3396  df-xp 4000  df-dm 4004

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