Theorem dminxp 3483

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 3305	. . . 4 |- dom ( C i^i (A X. B)) = ran `'(C i^i (A X. B))
2		cnvin 3456	. . . . . 6 |- `'(C i^i (A X. B)) = (`'C i^i `'(A X. B))
3		cnvxp 3464	. . . . . . 7 |- `'(A X. B) = (B X. A)
4	3	ineq2i 2214	. . . . . 6 |- (`'C i^i `'(A X. B)) = (`'C i^i (B X. A))
5	2, 4	eqtr 1495	. . . . 5 |- `'(C i^i (A X. B)) = (`'C i^i (B X. A))
6	5	rneqi 3340	. . . 4 |- ran `'(C i^i (A X. B)) = ran (`'C i^i (B X. A))
7	1, 6	eqtr 1495	. . 3 |- dom ( C i^i (A X. B)) = ran (`'C i^i (B X. A))
8	7	eqeq1i 1482	. 2 |- (dom ( C i^i (A X. B)) = A <-> ran (`'C i^i (B X. A)) = A)
9		rninxp 3482	. 2 |- (ran (`'C i^i (B X. A)) = A <-> A.x e. A E.y e. B y`'Cx)
10		visset 1813	. . . . 5 |- y e. V
11		visset 1813	. . . . 5 |- x e. V
12	10, 11	brcnv 3299	. . . 4 |- (y`'Cx <-> xCy)
13	12	rexbii 1668	. . 3 |- (E.y e. B y`'Cx <-> E.y e. B xCy)
14	13	ralbii 1667	. 2 |- (A.x e. A E.y e. B y`'Cx <-> A.x e. A E.y e. B xCy)
15	8, 9, 14	3bitr 177	1 |- (dom ( C i^i (A X. B)) = A <-> A.x e. A E.y e. B xCy)

Syntax hints:   <-> wb 146   = wceq 956  A.wral 1645  E.wrex 1646   i^i cin 2046   class class class wbr 2619   X. cxp 3168  `'ccnv 3169  dom cdm 3170  ran crn 3171

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-rex 1650  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-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190

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