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Theorem bnj900 28961
Description: Technical lemma for bnj69 29040. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj900.3  |-  D  =  ( om  \  { (/)
} )
bnj900.4  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
Assertion
Ref Expression
bnj900  |-  ( f  e.  B  ->  (/)  e.  dom  f )
Distinct variable group:    f, n
Allowed substitution hints:    ph( f, n)    ps( f, n)    B( f, n)    D( f, n)

Proof of Theorem bnj900
StepHypRef Expression
1 bnj900.4 . . . . . 6  |-  B  =  { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
21bnj1436 28872 . . . . 5  |-  ( f  e.  B  ->  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) )
3 simp1 955 . . . . . 6  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  ->  f  Fn  n )
43reximi 2650 . . . . 5  |-  ( E. n  e.  D  ( f  Fn  n  /\  ph 
/\  ps )  ->  E. n  e.  D  f  Fn  n )
5 fndm 5343 . . . . . 6  |-  ( f  Fn  n  ->  dom  f  =  n )
65reximi 2650 . . . . 5  |-  ( E. n  e.  D  f  Fn  n  ->  E. n  e.  D  dom  f  =  n )
72, 4, 63syl 18 . . . 4  |-  ( f  e.  B  ->  E. n  e.  D  dom  f  =  n )
87bnj1196 28827 . . 3  |-  ( f  e.  B  ->  E. n
( n  e.  D  /\  dom  f  =  n ) )
9 nfre1 2599 . . . . . . 7  |-  F/ n E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )
109nfab 2423 . . . . . 6  |-  F/_ n { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
111, 10nfcxfr 2416 . . . . 5  |-  F/_ n B
1211nfcri 2413 . . . 4  |-  F/ n  f  e.  B
131219.37 1809 . . 3  |-  ( E. n ( f  e.  B  ->  ( n  e.  D  /\  dom  f  =  n ) )  <->  ( f  e.  B  ->  E. n
( n  e.  D  /\  dom  f  =  n ) ) )
148, 13mpbir 200 . 2  |-  E. n
( f  e.  B  ->  ( n  e.  D  /\  dom  f  =  n ) )
15 nfv 1605 . . . 4  |-  F/ n (/) 
e.  dom  f
1612, 15nfim 1769 . . 3  |-  F/ n
( f  e.  B  -> 
(/)  e.  dom  f )
17 bnj900.3 . . . . . 6  |-  D  =  ( om  \  { (/)
} )
1817bnj529 28770 . . . . 5  |-  ( n  e.  D  ->  (/)  e.  n
)
19 eleq2 2344 . . . . . 6  |-  ( dom  f  =  n  -> 
( (/)  e.  dom  f  <->  (/)  e.  n ) )
2019biimparc 473 . . . . 5  |-  ( (
(/)  e.  n  /\  dom  f  =  n
)  ->  (/)  e.  dom  f )
2118, 20sylan 457 . . . 4  |-  ( ( n  e.  D  /\  dom  f  =  n
)  ->  (/)  e.  dom  f )
2221imim2i 13 . . 3  |-  ( ( f  e.  B  -> 
( n  e.  D  /\  dom  f  =  n ) )  ->  (
f  e.  B  ->  (/) 
e.  dom  f )
)
2316, 22exlimi 1801 . 2  |-  ( E. n ( f  e.  B  ->  ( n  e.  D  /\  dom  f  =  n ) )  -> 
( f  e.  B  -> 
(/)  e.  dom  f ) )
2414, 23ax-mp 8 1  |-  ( f  e.  B  ->  (/)  e.  dom  f )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544    \ cdif 3149   (/)c0 3455   {csn 3640   omcom 4656   dom cdm 4689    Fn wfn 5250
This theorem is referenced by:  bnj906  28962
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-13 1686  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-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-fn 5258
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