Theorem iuneq1 3834

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 3832	. . 3 |- ( A C_ B -> U_ x e. A C C_ U_ x e. B C )
2		iunss1 3832	. . 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 3117	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3117	. 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 1332   C_ wss 3076   U_ciun 3821

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3082  df-ss 3089  df-iun 3823

This theorem is referenced by:  iuneq1d  3844  iunxprg  3901  iununir  3904  iunsuc  4350  rdgisuc1  6289  rdg0  6292  oasuc  6368  omsuc  6376  iunfidisj  6842  fsum2d  11236  fsumiun  11278  iuncld  12323