MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunsuc Structured version   Visualization version   Unicode version

Theorem iunsuc 5508
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 5432 . . 3  |-  suc  A  =  ( A  u.  { A } )
2 iuneq1 4295 . . 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 4366 . 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 4364 . . 3  |-  U_ x  e.  { A } B  =  C
87uneq2i 3587 . 2  |-  ( U_ x  e.  A  B  u.  U_ x  e.  { A } B )  =  ( U_ x  e.  A  B  u.  C
)
93, 4, 83eqtri 2479 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 1446    e. wcel 1889   _Vcvv 3047    u. cun 3404   {csn 3970   U_ciun 4281   suc csuc 5428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ral 2744  df-rex 2745  df-v 3049  df-sbc 3270  df-un 3411  df-in 3413  df-ss 3420  df-sn 3971  df-iun 4283  df-suc 5432
This theorem is referenced by:  pwsdompw  8639
  Copyright terms: Public domain W3C validator