Theorem iuneq1 4020

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 4018	. . 3 |- ( A C_ B -> U_ x e. A C C_ U_ x e. B C )
2		iunss1 4018	. . 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 3263	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3263	. 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 1402   C_ wss 3220   U_ciun 4007

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-iun 4009

This theorem is referenced by:  iuneq1d  4030  iunxprg  4088  iununir  4091  iunsuc  4560  rdgisuc1  6645  rdg0  6648  oasuc  6727  omsuc  6735  iunfidisj  7250  fsum2d  12180  fsumiun  12222  fprod2d  12368  iuncld  15139