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Theorem iunsuc 4312
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 4263 . . 3  |-  suc  A  =  ( A  u.  { A } )
2 iuneq1 3796 . . 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 3862 . 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 3859 . . 3  |-  U_ x  e.  { A } B  =  C
87uneq2i 3197 . 2  |-  ( U_ x  e.  A  B  u.  U_ x  e.  { A } B )  =  ( U_ x  e.  A  B  u.  C
)
93, 4, 83eqtri 2142 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 1316    e. wcel 1465   _Vcvv 2660    u. cun 3039   {csn 3497   U_ciun 3783   suc csuc 4257
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-iun 3785  df-suc 4263
This theorem is referenced by: (None)
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