Theorem iuneq2 4537

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 4536	. . 3 |- ( A. x e. A B C_ C -> U_ x e. A B C_ U_ x e. A C )
2		ss2iun 4536	. . 3 |- ( A. x e. A C C_ B -> U_ x e. A C C_ U_ x e. A B )
3	1, 2	anim12i 590	. 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 3618	. . . 4 |- ( B = C <-> ( B C_ C /\ C C_ B ) )
5	4	ralbii 2980	. . 3 |- ( A. x e. A B = C <-> A. x e. A ( B C_ C /\ C C_ B ) )
6		r19.26 3064	. . 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 264	. 2 |- ( A. x e. A B = C <-> ( A. x e. A B C_ C /\ A. x e. A C C_ B ) )
8		eqss 3618	. 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 281	1 |- ( A. x e. A B = C -> U_ x e. A B = U_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   A.wral 2912   C_ wss 3574   U_ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-iun 4522

This theorem is referenced by:  iuneq2i  4539  iuneq2dv  4542  iunxdif3  4606  oa0r  7618  om0r  7619  om1r  7623  oe1m  7625  oaass  7641  oarec  7642  omass  7660  oeoalem  7676  oeoelem  7678  cardiun  8808  kmlem11  8982  iuncld  20849  comppfsc  21335  istotbnd3  33570  sstotbnd  33574  heibor  33620  iuneq12f  33972  cnvtrclfv  38016  iuneq2df  39212