Theorem iineq2 3274

Description: Equality theorem for indexed intersection. (The proof was shortened by Andrew Salmon, 25-Jul-2011.)

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		eleq2 1958	. . . . 5 |- (B = C -> (y e. B <-> y e. C))
2	1	ralimi 2168	. . . 4 |- (A.x e. A B = C -> A.x e. A (y e. B <-> y e. C))
3		ralbi 2223	. . . 4 |- (A.x e. A (y e. B <-> y e. C) -> (A.x e. A y e. B <-> A.x e. A y e. C))
4	2, 3	syl 12	. . 3 |- (A.x e. A B = C -> (A.x e. A y e. B <-> A.x e. A y e. C))
5	4	abbidv 2008	. 2 |- (A.x e. A B = C -> {y | A.x e. A y e. B} = {y | A.x e. A y e. C})
6		df-iin 3258	. 2 |- |^|_x e. A B = {y | A.x e. A y e. B}
7		df-iin 3258	. 2 |- |^|_x e. A C = {y | A.x e. A y e. C}
8	5, 6, 7	3eqtr4g 1953	1 |- (A.x e. A B = C -> |^|_x e. A B = |^|_x e. A C)

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  |^|_ciin 3256

This theorem is referenced by:  iineq2i 3277  iineq2d 3278  iincld 8955  sfcls 15604

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-10 1308  ax-12 1310  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

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-iin 3258

Copyright terms: Public domain