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Theorem bnj927 29139
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj927.1  |-  G  =  ( f  u.  { <. n ,  C >. } )
bnj927.2  |-  C  e. 
_V
Assertion
Ref Expression
bnj927  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p )

Proof of Theorem bnj927
StepHypRef Expression
1 simpr 448 . . . 4  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  f  Fn  n )
2 vex 2959 . . . . . 6  |-  n  e. 
_V
3 bnj927.2 . . . . . 6  |-  C  e. 
_V
42, 3fnsn 5504 . . . . 5  |-  { <. n ,  C >. }  Fn  { n }
54a1i 11 . . . 4  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  { <. n ,  C >. }  Fn  {
n } )
6 bnj521 29104 . . . . 5  |-  ( n  i^i  { n }
)  =  (/)
76a1i 11 . . . 4  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  ( n  i^i  { n } )  =  (/) )
8 fnun 5551 . . . 4  |-  ( ( ( f  Fn  n  /\  { <. n ,  C >. }  Fn  { n } )  /\  (
n  i^i  { n } )  =  (/) )  ->  ( f  u. 
{ <. n ,  C >. } )  Fn  (
n  u.  { n } ) )
91, 5, 7, 8syl21anc 1183 . . 3  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  ( f  u.  { <. n ,  C >. } )  Fn  (
n  u.  { n } ) )
10 bnj927.1 . . . 4  |-  G  =  ( f  u.  { <. n ,  C >. } )
1110fneq1i 5539 . . 3  |-  ( G  Fn  ( n  u. 
{ n } )  <-> 
( f  u.  { <. n ,  C >. } )  Fn  ( n  u.  { n }
) )
129, 11sylibr 204 . 2  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  ( n  u.  { n } ) )
13 df-suc 4587 . . . . . 6  |-  suc  n  =  ( n  u. 
{ n } )
1413eqeq2i 2446 . . . . 5  |-  ( p  =  suc  n  <->  p  =  ( n  u.  { n } ) )
1514biimpi 187 . . . 4  |-  ( p  =  suc  n  ->  p  =  ( n  u.  { n } ) )
1615adantr 452 . . 3  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  p  =  ( n  u.  { n } ) )
1716fneq2d 5537 . 2  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  ( G  Fn  p  <->  G  Fn  (
n  u.  { n } ) ) )
1812, 17mpbird 224 1  |-  ( ( 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    i^i cin 3319   (/)c0 3628   {csn 3814   <.cop 3817   suc csuc 4583    Fn wfn 5449
This theorem is referenced by:  bnj941  29143  bnj929  29307
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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