Theorem iuneq1 3978

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 3976	. . 3 |- ( A C_ B -> U_ x e. A C C_ U_ x e. B C )
2		iunss1 3976	. . 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 3239	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3239	. 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 1395   C_ wss 3197   U_ciun 3965

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-iun 3967

This theorem is referenced by:  iuneq1d  3988  iunxprg  4046  iununir  4049  iunsuc  4511  rdgisuc1  6530  rdg0  6533  oasuc  6610  omsuc  6618  iunfidisj  7113  fsum2d  11946  fsumiun  11988  fprod2d  12134  iuncld  14789