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Theorem bnj900 29740
Description: Technical lemma for bnj69 29819. 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 29651 . . . . 5  |-  ( f  e.  B  ->  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) )
3 simp1 1008 . . . . . 6  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  ->  f  Fn  n )
43reximi 2855 . . . . 5  |-  ( E. n  e.  D  ( f  Fn  n  /\  ph 
/\  ps )  ->  E. n  e.  D  f  Fn  n )
5 fndm 5675 . . . . . 6  |-  ( f  Fn  n  ->  dom  f  =  n )
65reximi 2855 . . . . 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 29606 . . 3  |-  ( f  e.  B  ->  E. n
( n  e.  D  /\  dom  f  =  n ) )
9 nfre1 2848 . . . . . . 7  |-  F/ n E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )
109nfab 2596 . . . . . 6  |-  F/_ n { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
111, 10nfcxfr 2590 . . . . 5  |-  F/_ n B
1211nfcri 2586 . . . 4  |-  F/ n  f  e.  B
131219.37 2046 . . 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 213 . 2  |-  E. n
( f  e.  B  ->  ( n  e.  D  /\  dom  f  =  n ) )
15 nfv 1761 . . . 4  |-  F/ n (/) 
e.  dom  f
1612, 15nfim 2003 . . 3  |-  F/ n
( f  e.  B  -> 
(/)  e.  dom  f )
17 bnj900.3 . . . . . 6  |-  D  =  ( om  \  { (/)
} )
1817bnj529 29551 . . . . 5  |-  ( n  e.  D  ->  (/)  e.  n
)
19 eleq2 2518 . . . . . 6  |-  ( dom  f  =  n  -> 
( (/)  e.  dom  f  <->  (/)  e.  n ) )
2019biimparc 490 . . . . 5  |-  ( (
(/)  e.  n  /\  dom  f  =  n
)  ->  (/)  e.  dom  f )
2118, 20sylan 474 . . . 4  |-  ( ( n  e.  D  /\  dom  f  =  n
)  ->  (/)  e.  dom  f )
2221imim2i 16 . . 3  |-  ( ( f  e.  B  -> 
( n  e.  D  /\  dom  f  =  n ) )  ->  (
f  e.  B  ->  (/) 
e.  dom  f )
)
2316, 22exlimi 1995 . 2  |-  ( E. n ( f  e.  B  ->  ( n  e.  D  /\  dom  f  =  n ) )  -> 
( f  e.  B  -> 
(/)  e.  dom  f ) )
2414, 23ax-mp 5 1  |-  ( f  e.  B  ->  (/)  e.  dom  f )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444   E.wex 1663    e. wcel 1887   {cab 2437   E.wrex 2738    \ cdif 3401   (/)c0 3731   {csn 3968   dom cdm 4834    Fn wfn 5577   omcom 6692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-tr 4498  df-eprel 4745  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-fn 5585  df-om 6693
This theorem is referenced by:  bnj906  29741
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