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Theorem bnj941 29143
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 3985 . . . . . . 7  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  <. n ,  C >.  =  <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. )
32sneqd 3827 . . . . . 6  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  { <. n ,  C >. }  =  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )
43uneq2d 3501 . . . . 5  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  (
f  u.  { <. n ,  C >. } )  =  ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } ) )
51, 4syl5eq 2480 . . . 4  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  G  =  ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } ) )
65fneq1d 5536 . . 3  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  ( G  Fn  p  <->  ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  Fn  p
) )
76imbi2d 308 . 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 2436 . . 3  |-  ( f  u.  { <. n ,  if ( C  e. 
_V ,  C ,  (/) ) >. } )  =  ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )
9 0ex 4339 . . . 4  |-  (/)  e.  _V
109elimel 3791 . . 3  |-  if ( C  e.  _V ,  C ,  (/) )  e. 
_V
118, 10bnj927 29139 . 2  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  Fn  p
)
127, 11dedth 3780 1  |-  ( C  e.  _V  ->  (
( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    u. cun 3318   (/)c0 3628   ifcif 3739   {csn 3814   <.cop 3817   suc csuc 4583    Fn wfn 5449
This theorem is referenced by:  bnj945  29144  bnj910  29319
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-reg 7560
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-fun 5456  df-fn 5457
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