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Theorem iunsuc 5491
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
StepHypRef Expression
1 df-suc 5415 . . 3  |-  suc  A  =  ( A  u.  { A } )
2 iuneq1 4284 . . 3  |-  ( suc 
A  =  ( A  u.  { A }
)  ->  U_ x  e. 
suc  A B  = 
U_ x  e.  ( A  u.  { A } ) B )
31, 2ax-mp 5 . 2  |-  U_ x  e.  suc  A B  = 
U_ x  e.  ( A  u.  { A } ) B
4 iunxun 4355 . 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 )
75, 6iunxsn 4353 . . 3  |-  U_ x  e.  { A } B  =  C
87uneq2i 3593 . 2  |-  ( U_ x  e.  A  B  u.  U_ x  e.  { A } B )  =  ( U_ x  e.  A  B  u.  C
)
93, 4, 83eqtri 2435 1  |-  U_ x  e.  suc  A B  =  ( U_ x  e.  A  B  u.  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   _Vcvv 3058    u. cun 3411   {csn 3971   U_ciun 4270   suc csuc 5411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-v 3060  df-sbc 3277  df-un 3418  df-in 3420  df-ss 3427  df-sn 3972  df-iun 4272  df-suc 5415
This theorem is referenced by:  pwsdompw  8615
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