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Theorem cbviun 3955
Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)
Hypotheses
Ref Expression
cbviun.1  |-  F/_ y B
cbviun.2  |-  F/_ x C
cbviun.3  |-  ( x  =  y  ->  B  =  C )
Assertion
Ref Expression
cbviun  |-  U_ x  e.  A  B  =  U_ y  e.  A  C
Distinct variable groups:    y, A    x, A
Allowed substitution hints:    B( x, y)    C( x, y)

Proof of Theorem cbviun
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cbviun.1 . . . . 5  |-  F/_ y B
21nfcri 2426 . . . 4  |-  F/ y  z  e.  B
3 cbviun.2 . . . . 5  |-  F/_ x C
43nfcri 2426 . . . 4  |-  F/ x  z  e.  C
5 cbviun.3 . . . . 5  |-  ( x  =  y  ->  B  =  C )
65eleq2d 2363 . . . 4  |-  ( x  =  y  ->  (
z  e.  B  <->  z  e.  C ) )
72, 4, 6cbvrex 2774 . . 3  |-  ( E. x  e.  A  z  e.  B  <->  E. y  e.  A  z  e.  C )
87abbii 2408 . 2  |-  { z  |  E. x  e.  A  z  e.  B }  =  { z  |  E. y  e.  A  z  e.  C }
9 df-iun 3923 . 2  |-  U_ x  e.  A  B  =  { z  |  E. x  e.  A  z  e.  B }
10 df-iun 3923 . 2  |-  U_ y  e.  A  C  =  { z  |  E. y  e.  A  z  e.  C }
118, 9, 103eqtr4i 2326 1  |-  U_ x  e.  A  B  =  U_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {cab 2282   F/_wnfc 2419   E.wrex 2557   U_ciun 3921
This theorem is referenced by:  cbviunv  3957  disjxiun  4036  funiunfvf  5791  mpt2mptsx  6203  dmmpt2ssx  6205  fmpt2x  6206  ovmptss  6216  iunfi  7160  fsum2dlem  12249  fsumcom2  12253  fsumiun  12295  gsumcom2  15242  fiuncmp  17147  ovolfiniun  18876  ovoliunlem3  18879  ovoliun  18880  finiunmbl  18917  volfiniun  18920  iunmbl  18926  limciun  19260  hashunif  23400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-iun 3923
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