Theorem iunsuc 5807

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 5729	. . 3 |- suc A = ( A u. { A } )
2		iuneq1 4534	. . 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 4605	. 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 4603	. . 3 |- U_ x e. { A } B = C
8	7	uneq2i 3764	. 2 |- ( U_ x e. A B u. U_ x e. { A } B ) = ( U_ x e. A B u. C )
9	3, 4, 8	3eqtri 2648	1 |- U_ x e. suc A B = ( U_ x e. A B u. C )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   {csn 4177   U_ciun 4520   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-iun 4522  df-suc 5729

This theorem is referenced by:  pwsdompw  9026