MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iuneq2 Structured version   Unicode version

Theorem iuneq2 4290
Description: Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)
Assertion
Ref Expression
iuneq2  |-  ( A. x  e.  A  B  =  C  ->  U_ x  e.  A  B  =  U_ x  e.  A  C
)

Proof of Theorem iuneq2
StepHypRef Expression
1 ss2iun 4289 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  U_ x  e.  A  B  C_  U_ x  e.  A  C )
2 ss2iun 4289 . . 3  |-  ( A. x  e.  A  C  C_  B  ->  U_ x  e.  A  C  C_  U_ x  e.  A  B )
31, 2anim12i 566 . 2  |-  ( ( A. x  e.  A  B  C_  C  /\  A. x  e.  A  C  C_  B )  ->  ( U_ x  e.  A  B  C_  U_ x  e.  A  C  /\  U_ x  e.  A  C  C_ 
U_ x  e.  A  B ) )
4 eqss 3474 . . . 4  |-  ( B  =  C  <->  ( B  C_  C  /\  C  C_  B ) )
54ralbii 2836 . . 3  |-  ( A. x  e.  A  B  =  C  <->  A. x  e.  A  ( B  C_  C  /\  C  C_  B ) )
6 r19.26 2949 . . 3  |-  ( A. x  e.  A  ( B  C_  C  /\  C  C_  B )  <->  ( A. x  e.  A  B  C_  C  /\  A. x  e.  A  C  C_  B
) )
75, 6bitri 249 . 2  |-  ( A. x  e.  A  B  =  C  <->  ( A. x  e.  A  B  C_  C  /\  A. x  e.  A  C  C_  B ) )
8 eqss 3474 . 2  |-  ( U_ x  e.  A  B  =  U_ x  e.  A  C 
<->  ( U_ x  e.  A  B  C_  U_ x  e.  A  C  /\  U_ x  e.  A  C  C_ 
U_ x  e.  A  B ) )
93, 7, 83imtr4i 266 1  |-  ( A. x  e.  A  B  =  C  ->  U_ x  e.  A  B  =  U_ x  e.  A  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370   A.wral 2796    C_ wss 3431   U_ciun 4274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ral 2801  df-rex 2802  df-v 3074  df-in 3438  df-ss 3445  df-iun 4276
This theorem is referenced by:  iuneq2i  4292  iuneq2dv  4295  oa0r  7083  om0r  7084  om1r  7087  oe1m  7089  oaass  7105  oarec  7106  omass  7124  oeoalem  7140  oeoelem  7142  cardiun  8258  kmlem11  8435  iuncld  18776  ofpreima2  26131  comppfsc  28722  istotbnd3  28813  sstotbnd  28817  heibor  28863  iuneq12f  29119
  Copyright terms: Public domain W3C validator