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Theorem iuneq2 4101
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 4100 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  U_ x  e.  A  B  C_  U_ x  e.  A  C )
2 ss2iun 4100 . . 3  |-  ( A. x  e.  A  C  C_  B  ->  U_ x  e.  A  C  C_  U_ x  e.  A  B )
31, 2anim12i 550 . 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 3355 . . . 4  |-  ( B  =  C  <->  ( B  C_  C  /\  C  C_  B ) )
54ralbii 2721 . . 3  |-  ( A. x  e.  A  B  =  C  <->  A. x  e.  A  ( B  C_  C  /\  C  C_  B ) )
6 r19.26 2830 . . 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 241 . 2  |-  ( A. x  e.  A  B  =  C  <->  ( A. x  e.  A  B  C_  C  /\  A. x  e.  A  C  C_  B ) )
8 eqss 3355 . 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 258 1  |-  ( A. x  e.  A  B  =  C  ->  U_ x  e.  A  B  =  U_ x  e.  A  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652   A.wral 2697    C_ wss 3312   U_ciun 4085
This theorem is referenced by:  iuneq2i  4103  iuneq2dv  4106  abianfplem  6707  oa0r  6774  om0r  6775  om1r  6778  oe1m  6780  oaass  6796  oarec  6797  omass  6815  oeoalem  6831  oeoelem  6833  cardiun  7861  kmlem11  8032  iuncld  17101  sibfof  24646  comppfsc  26378  istotbnd3  26471  sstotbnd  26475  heibor  26521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-in 3319  df-ss 3326  df-iun 4087
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