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Theorem iunsuc 4474
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 4398 . . 3  |-  suc  A  =  ( A  u.  { A } )
2 iuneq1 3918 . . 3  |-  ( suc 
A  =  ( A  u.  { A }
)  ->  U_ x  e. 
suc  A B  = 
U_ x  e.  ( A  u.  { A } ) B )
31, 2ax-mp 8 . 2  |-  U_ x  e.  suc  A B  = 
U_ x  e.  ( A  u.  { A } ) B
4 iunxun 3983 . 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 3981 . . 3  |-  U_ x  e.  { A } B  =  C
87uneq2i 3326 . 2  |-  ( U_ x  e.  A  B  u.  U_ x  e.  { A } B )  =  ( U_ x  e.  A  B  u.  C
)
93, 4, 83eqtri 2307 1  |-  U_ x  e.  suc  A B  =  ( U_ x  e.  A  B  u.  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150   {csn 3640   U_ciun 3905   suc csuc 4394
This theorem is referenced by:  pwsdompw  7830
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992  df-un 3157  df-in 3159  df-ss 3166  df-sn 3646  df-iun 3907  df-suc 4398
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