Theorem iuneq2 4290

Description: Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)

Assertion

Ref	Expression
iuneq2	|- ( A. x e. A B = C -> U_ x e. A B = U_ x e. A C )

Proof of Theorem iuneq2

Step	Hyp	Ref	Expression
1		ss2iun 4289	. . 3 |- ( A. x e. A B C_ C -> U_ x e. A B C_ U_ x e. A C )
2		ss2iun 4289	. . 3 |- ( A. x e. A C C_ B -> U_ x e. A C C_ U_ x e. A B )
3	1, 2	anim12i 566	. 2 |- ( ( A. x e. A B C_ C /\ A. x e. A C C_ B ) -> ( U_ x e. A B C_ U_ x e. A C /\ U_ x e. A C C_ U_ x e. A B ) )
4		eqss 3474	. . . 4 |- ( B = C <-> ( B C_ C /\ C C_ B ) )
5	4	ralbii 2836	. . 3 |- ( A. x e. A B = C <-> A. x e. A ( B C_ C /\ C C_ B ) )
6		r19.26 2949	. . 3 |- ( A. x e. A ( B C_ C /\ C C_ B ) <-> ( A. x e. A B C_ C /\ A. x e. A C C_ B ) )
7	5, 6	bitri 249	. 2 |- ( A. x e. A B = C <-> ( A. x e. A B C_ C /\ A. x e. A C C_ B ) )
8		eqss 3474	. 2 |- ( U_ x e. A B = U_ x e. A C <-> ( U_ x e. A B C_ U_ x e. A C /\ U_ x e. A C C_ U_ x e. A B ) )
9	3, 7, 8	3imtr4i 266	1 |- ( A. x e. A B = C -> U_ x e. A B = U_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   A.wral 2796   C_ wss 3431   U_ciun 4274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ral 2801  df-rex 2802  df-v 3074  df-in 3438  df-ss 3445  df-iun 4276

This theorem is referenced by:  iuneq2i  4292  iuneq2dv  4295  oa0r  7083  om0r  7084  om1r  7087  oe1m  7089  oaass  7105  oarec  7106  omass  7124  oeoalem  7140  oeoelem  7142  cardiun  8258  kmlem11  8435  iuncld  18776  ofpreima2  26131  comppfsc  28722  istotbnd3  28813  sstotbnd  28817  heibor  28863  iuneq12f  29119