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Theorem bnj941 29120
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj941.1  |-  G  =  ( f  u.  { <. n ,  C >. } )
Assertion
Ref Expression
bnj941  |-  ( C  e.  _V  ->  (
( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p ) )

Proof of Theorem bnj941
StepHypRef Expression
1 bnj941.1 . . . . 5  |-  G  =  ( f  u.  { <. n ,  C >. } )
2 opeq2 3813 . . . . . . 7  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  <. n ,  C >.  =  <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. )
32sneqd 3666 . . . . . 6  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  { <. n ,  C >. }  =  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )
43uneq2d 3342 . . . . 5  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  (
f  u.  { <. n ,  C >. } )  =  ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } ) )
51, 4syl5eq 2340 . . . 4  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  G  =  ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } ) )
65fneq1d 5351 . . 3  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  ( G  Fn  p  <->  ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  Fn  p
) )
76imbi2d 307 . 2  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  (
( ( p  =  suc  n  /\  f  Fn  n )  ->  G  Fn  p )  <->  ( (
p  =  suc  n  /\  f  Fn  n
)  ->  ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  Fn  p
) ) )
8 eqid 2296 . . 3  |-  ( f  u.  { <. n ,  if ( C  e. 
_V ,  C ,  (/) ) >. } )  =  ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )
9 0ex 4166 . . . 4  |-  (/)  e.  _V
109elimel 3630 . . 3  |-  if ( C  e.  _V ,  C ,  (/) )  e. 
_V
118, 10bnj927 29116 . 2  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  Fn  p
)
127, 11dedth 3619 1  |-  ( C  e.  _V  ->  (
( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163   (/)c0 3468   ifcif 3578   {csn 3653   <.cop 3656   suc csuc 4410    Fn wfn 5266
This theorem is referenced by:  bnj945  29121  bnj910  29296
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-reg 7322
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-fun 5273  df-fn 5274
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