Theorem iineq2 2007

Description: Equality theorem for indexed intersection.

Assertion

Ref	Expression
iineq2	|- (A.x e. A B = C -> |^|x e. A B = |^|x e. A C)

Proof of Theorem iineq2

Step	Hyp	Ref	Expression
1		hbra1 1237	. . . 4 |- (A.x e. A B = C -> A.xA.x e. A B = C)
2		ra4 1243	. . . . . 6 |- (A.x e. A B = C -> (x e. A -> B = C))
3		eleq2 1150	. . . . . 6 |- (B = C -> (y e. B <-> y e. C))
4	2, 3	syl6 23	. . . . 5 |- (A.x e. A B = C -> (x e. A -> (y e. B <-> y e. C)))
5	4	imp 277	. . . 4 |- ((A.x e. A B = C /\ x e. A) -> (y e. B <-> y e. C))
6	1, 5	biralda 1213	. . 3 |- (A.x e. A B = C -> (A.x e. A y e. B <-> A.x e. A y e. C))
7	6	biabdv 1183	. 2 |- (A.x e. A B = C -> {y | A.x e. A y e. B} = {y | A.x e. A y e. C})
8		df-iin 1997	. 2 |- |^|x e. A B = {y | A.x e. A y e. B}
9		df-iin 1997	. 2 |- |^|x e. A C = {y | A.x e. A y e. C}
10	7, 8, 9	3eqtr4g 1147	1 |- (A.x e. A B = C -> |^|x e. A B = |^|x e. A C)

Syntax hints:   -> wi 2   <-> wb 127  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  |^|ciin 1995

This theorem is referenced by:  iineq2i 2009  iineq2dv 2011

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-iin 1997

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