Theorem dminxp 4357

Description: Domain of the intersection with a cross product.

Assertion

Ref	Expression
dminxp	|- (dom ( C i^i (A X. B)) = A <-> A.x e. A E.y e. B xCy)

Distinct variable groups:   x,A   x,y,B   x,C,y

Proof of Theorem dminxp

Step	Hyp	Ref	Expression
1		dfdm4 4151	. . . 4 |- dom ( C i^i (A X. B)) = ran `'(C i^i (A X. B))
2		cnvin 4324	. . . . . 6 |- `'(C i^i (A X. B)) = (`'C i^i `'(A X. B))
3		cnvxp 4332	. . . . . . 7 |- `'(A X. B) = (B X. A)
4	3	ineq2i 2793	. . . . . 6 |- (`'C i^i `'(A X. B)) = (`'C i^i (B X. A))
5	2, 4	eqtri 1908	. . . . 5 |- `'(C i^i (A X. B)) = (`'C i^i (B X. A))
6	5	rneqi 4187	. . . 4 |- ran `'(C i^i (A X. B)) = ran (`'C i^i (B X. A))
7	1, 6	eqtri 1908	. . 3 |- dom ( C i^i (A X. B)) = ran (`'C i^i (B X. A))
8	7	eqeq1i 1891	. 2 |- (dom ( C i^i (A X. B)) = A <-> ran (`'C i^i (B X. A)) = A)
9		rninxp 4355	. 2 |- (ran (`'C i^i (B X. A)) = A <-> A.x e. A E.y e. B y`'Cx)
10		visset 2295	. . . . 5 |- y e. _V
11		visset 2295	. . . . 5 |- x e. _V
12	10, 11	brcnv 4144	. . . 4 |- (y`'Cx <-> xCy)
13	12	rexbii 2128	. . 3 |- (E.y e. B y`'Cx <-> E.y e. B xCy)
14	13	ralbii 2127	. 2 |- (A.x e. A E.y e. B y`'Cx <-> A.x e. A E.y e. B xCy)
15	8, 9, 14	3bitri 194	1 |- (dom ( C i^i (A X. B)) = A <-> A.x e. A E.y e. B xCy)

Syntax hints:   <-> wb 163   = wceq 1298  A.wral 2105  E.wrex 2106   i^i cin 2592   class class class wbr 3338   X. cxp 3984  `'ccnv 3985  dom cdm 3986  ran crn 3987

This theorem is referenced by:  rngsubpos 14636

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-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007

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