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Theorem bnj941 28849
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 3945 . . . . . . 7  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  <. n ,  C >.  =  <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. )
32sneqd 3787 . . . . . 6  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  { <. n ,  C >. }  =  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )
43uneq2d 3461 . . . . 5  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  (
f  u.  { <. n ,  C >. } )  =  ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } ) )
51, 4syl5eq 2448 . . . 4  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  G  =  ( f  u. 
{ <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } ) )
65fneq1d 5495 . . 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 2404 . . 3  |-  ( f  u.  { <. n ,  if ( C  e. 
_V ,  C ,  (/) ) >. } )  =  ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )
9 0ex 4299 . . . 4  |-  (/)  e.  _V
109elimel 3751 . . 3  |-  if ( C  e.  _V ,  C ,  (/) )  e. 
_V
118, 10bnj927 28845 . 2  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  ( f  u.  { <. n ,  if ( C  e.  _V ,  C ,  (/) ) >. } )  Fn  p
)
127, 11dedth 3740 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 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278   (/)c0 3588   ifcif 3699   {csn 3774   <.cop 3777   suc csuc 4543    Fn wfn 5408
This theorem is referenced by:  bnj945  28850  bnj910  29025
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-reg 7516
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-id 4458  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-fun 5415  df-fn 5416
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