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Theorem bnj927 28800
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 447 . . . 4  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  f  Fn  n )
2 vex 2791 . . . . . 6  |-  n  e. 
_V
3 bnj927.2 . . . . . 6  |-  C  e. 
_V
42, 3fnsn 5304 . . . . 5  |-  { <. n ,  C >. }  Fn  { n }
54a1i 10 . . . 4  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  { <. n ,  C >. }  Fn  {
n } )
6 bnj521 28765 . . . . 5  |-  ( n  i^i  { n }
)  =  (/)
76a1i 10 . . . 4  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  ( n  i^i  { n } )  =  (/) )
8 fnun 5350 . . . 4  |-  ( ( ( f  Fn  n  /\  { <. n ,  C >. }  Fn  { n } )  /\  (
n  i^i  { n } )  =  (/) )  ->  ( f  u. 
{ <. n ,  C >. } )  Fn  (
n  u.  { n } ) )
91, 5, 7, 8syl21anc 1181 . . 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 5338 . . 3  |-  ( G  Fn  ( n  u. 
{ n } )  <-> 
( f  u.  { <. n ,  C >. } )  Fn  ( n  u.  { n }
) )
129, 11sylibr 203 . 2  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  ( n  u.  { n } ) )
13 df-suc 4398 . . . . . 6  |-  suc  n  =  ( n  u. 
{ n } )
1413eqeq2i 2293 . . . . 5  |-  ( p  =  suc  n  <->  p  =  ( n  u.  { n } ) )
1514biimpi 186 . . . 4  |-  ( p  =  suc  n  ->  p  =  ( n  u.  { n } ) )
1615adantr 451 . . 3  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  p  =  ( n  u.  { n } ) )
1716fneq2d 5336 . 2  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  ( G  Fn  p  <->  G  Fn  (
n  u.  { n } ) ) )
1812, 17mpbird 223 1  |-  ( ( p  =  suc  n  /\  f  Fn  n
)  ->  G  Fn  p )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150    i^i cin 3151   (/)c0 3455   {csn 3640   <.cop 3643   suc csuc 4394    Fn wfn 5250
This theorem is referenced by:  bnj941  28804  bnj929  28968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-reg 7306
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257  df-fn 5258
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