Theorem iuneq1 3901

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 3899	. . 3 |- ( A C_ B -> U_ x e. A C C_ U_ x e. B C )
2		iunss1 3899	. . 3 |- ( B C_ A -> U_ x e. B C C_ U_ x e. A C )
3	1, 2	anim12i 338	. 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 3172	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3172	. 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 201	1 |- ( A = B -> U_ x e. A C = U_ x e. B C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   C_ wss 3131   U_ciun 3888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-iun 3890

This theorem is referenced by:  iuneq1d  3911  iunxprg  3969  iununir  3972  iunsuc  4422  rdgisuc1  6387  rdg0  6390  oasuc  6467  omsuc  6475  iunfidisj  6947  fsum2d  11445  fsumiun  11487  fprod2d  11633  iuncld  13700