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Theorem bnj900 32274
Description: Technical lemma for bnj69 32353. 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 32185 . . . . 5  |-  ( f  e.  B  ->  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) )
3 simp1 988 . . . . . 6  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  ->  f  Fn  n )
43reximi 2929 . . . . 5  |-  ( E. n  e.  D  ( f  Fn  n  /\  ph 
/\  ps )  ->  E. n  e.  D  f  Fn  n )
5 fndm 5621 . . . . . 6  |-  ( f  Fn  n  ->  dom  f  =  n )
65reximi 2929 . . . . 5  |-  ( E. n  e.  D  f  Fn  n  ->  E. n  e.  D  dom  f  =  n )
72, 4, 63syl 20 . . . 4  |-  ( f  e.  B  ->  E. n  e.  D  dom  f  =  n )
87bnj1196 32140 . . 3  |-  ( f  e.  B  ->  E. n
( n  e.  D  /\  dom  f  =  n ) )
9 nfre1 2891 . . . . . . 7  |-  F/ n E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )
109nfab 2620 . . . . . 6  |-  F/_ n { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
111, 10nfcxfr 2614 . . . . 5  |-  F/_ n B
1211nfcri 2609 . . . 4  |-  F/ n  f  e.  B
131219.37 1906 . . 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 209 . 2  |-  E. n
( f  e.  B  ->  ( n  e.  D  /\  dom  f  =  n ) )
15 nfv 1674 . . . 4  |-  F/ n (/) 
e.  dom  f
1612, 15nfim 1858 . . 3  |-  F/ n
( f  e.  B  -> 
(/)  e.  dom  f )
17 bnj900.3 . . . . . 6  |-  D  =  ( om  \  { (/)
} )
1817bnj529 32085 . . . . 5  |-  ( n  e.  D  ->  (/)  e.  n
)
19 eleq2 2527 . . . . . 6  |-  ( dom  f  =  n  -> 
( (/)  e.  dom  f  <->  (/)  e.  n ) )
2019biimparc 487 . . . . 5  |-  ( (
(/)  e.  n  /\  dom  f  =  n
)  ->  (/)  e.  dom  f )
2118, 20sylan 471 . . . 4  |-  ( ( n  e.  D  /\  dom  f  =  n
)  ->  (/)  e.  dom  f )
2221imim2i 14 . . 3  |-  ( ( f  e.  B  -> 
( n  e.  D  /\  dom  f  =  n ) )  ->  (
f  e.  B  ->  (/) 
e.  dom  f )
)
2316, 22exlimi 1850 . 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 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2439   E.wrex 2800    \ cdif 3436   (/)c0 3748   {csn 3988   dom cdm 4951    Fn wfn 5524   omcom 6589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-tr 4497  df-eprel 4743  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-fn 5532  df-om 6590
This theorem is referenced by:  bnj906  32275
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