Theorem iuneq1 3879

Description: Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)

Assertion

Ref	Expression
iuneq1	|- ( A = B -> U_ x e. A C = U_ x e. B C )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem iuneq1

Step	Hyp	Ref	Expression
1		iunss1 3877	. . 3 |- ( A C_ B -> U_ x e. A C C_ U_ x e. B C )
2		iunss1 3877	. . 3 |- ( B C_ A -> U_ x e. B C C_ U_ x e. A C )
3	1, 2	anim12i 336	. 2 |- ( ( A C_ B /\ B C_ A ) -> ( U_ x e. A C C_ U_ x e. B C /\ U_ x e. B C C_ U_ x e. A C ) )
4		eqss 3157	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3157	. 2 |- ( U_ x e. A C = U_ x e. B C <-> ( U_ x e. A C C_ U_ x e. B C /\ U_ x e. B C C_ U_ x e. A C ) )
6	3, 4, 5	3imtr4i 200	1 |- ( A = B -> U_ x e. A C = U_ x e. B C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   C_ wss 3116   U_ciun 3866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-iun 3868

This theorem is referenced by:  iuneq1d  3889  iunxprg  3946  iununir  3949  iunsuc  4398  rdgisuc1  6352  rdg0  6355  oasuc  6432  omsuc  6440  iunfidisj  6911  fsum2d  11376  fsumiun  11418  fprod2d  11564  iuncld  12755