Theorem iuneq2 3986

Description: Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)

Assertion

Ref	Expression
iuneq2	|- ( A. x e. A B = C -> U_ x e. A B = U_ x e. A C )

Proof of Theorem iuneq2

Step	Hyp	Ref	Expression
1		ss2iun 3985	. . 3 |- ( A. x e. A B C_ C -> U_ x e. A B C_ U_ x e. A C )
2		ss2iun 3985	. . 3 |- ( A. x e. A C C_ B -> U_ x e. A C C_ U_ x e. A B )
3	1, 2	anim12i 338	. 2 |- ( ( A. x e. A B C_ C /\ A. x e. A C C_ B ) -> ( U_ x e. A B C_ U_ x e. A C /\ U_ x e. A C C_ U_ x e. A B ) )
4		eqss 3242	. . . 4 |- ( B = C <-> ( B C_ C /\ C C_ B ) )
5	4	ralbii 2538	. . 3 |- ( A. x e. A B = C <-> A. x e. A ( B C_ C /\ C C_ B ) )
6		r19.26 2659	. . 3 |- ( A. x e. A ( B C_ C /\ C C_ B ) <-> ( A. x e. A B C_ C /\ A. x e. A C C_ B ) )
7	5, 6	bitri 184	. 2 |- ( A. x e. A B = C <-> ( A. x e. A B C_ C /\ A. x e. A C C_ B ) )
8		eqss 3242	. 2 |- ( U_ x e. A B = U_ x e. A C <-> ( U_ x e. A B C_ U_ x e. A C /\ U_ x e. A C C_ U_ x e. A B ) )
9	3, 7, 8	3imtr4i 201	1 |- ( A. x e. A B = C -> U_ x e. A B = U_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   A.wral 2510   C_ wss 3200   U_ciun 3970

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-iun 3972

This theorem is referenced by:  iuneq2i  3988  iuneq2dv  3991  dfmptg  5826