Theorem iunsuc 4337

Description: Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Hypotheses

Ref	Expression
iunsuc.1	|- A e. _V
iunsuc.2	|- ( x = A -> B = C )

Assertion

Ref	Expression
iunsuc	|- U_ x e. suc A B = ( U_ x e. A B u. C )

Distinct variable groups:   x, A   x, C

Allowed substitution hint:   B( x)

Proof of Theorem iunsuc

Step	Hyp	Ref	Expression
1		df-suc 4288	. . 3 |- suc A = ( A u. { A } )
2		iuneq1 3821	. . 3 |- ( suc A = ( A u. { A } ) -> U_ x e. suc A B = U_ x e. ( A u. { A } ) B )
3	1, 2	ax-mp 5	. 2 |- U_ x e. suc A B = U_ x e. ( A u. { A } ) B
4		iunxun 3887	. 2 |- U_ x e. ( A u. { A } ) B = ( U_ x e. A B u. U_ x e. { A } B )
5		iunsuc.1	. . . 4 |- A e. _V
6		iunsuc.2	. . . 4 |- ( x = A -> B = C )
7	5, 6	iunxsn 3884	. . 3 |- U_ x e. { A } B = C
8	7	uneq2i 3222	. 2 |- ( U_ x e. A B u. U_ x e. { A } B ) = ( U_ x e. A B u. C )
9	3, 4, 8	3eqtri 2162	1 |- U_ x e. suc A B = ( U_ x e. A B u. C )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   _Vcvv 2681   u. cun 3064   {csn 3522   U_ciun 3808   suc csuc 4282

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-iun 3810  df-suc 4288

This theorem is referenced by: (None)