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Theorem bnj900 28639
Description: Technical lemma for bnj69 28718. 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 28550 . . . . 5  |-  ( f  e.  B  ->  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) )
3 simp1 957 . . . . . 6  |-  ( ( f  Fn  n  /\  ph 
/\  ps )  ->  f  Fn  n )
43reximi 2757 . . . . 5  |-  ( E. n  e.  D  ( f  Fn  n  /\  ph 
/\  ps )  ->  E. n  e.  D  f  Fn  n )
5 fndm 5485 . . . . . 6  |-  ( f  Fn  n  ->  dom  f  =  n )
65reximi 2757 . . . . 5  |-  ( E. n  e.  D  f  Fn  n  ->  E. n  e.  D  dom  f  =  n )
72, 4, 63syl 19 . . . 4  |-  ( f  e.  B  ->  E. n  e.  D  dom  f  =  n )
87bnj1196 28505 . . 3  |-  ( f  e.  B  ->  E. n
( n  e.  D  /\  dom  f  =  n ) )
9 nfre1 2706 . . . . . . 7  |-  F/ n E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps )
109nfab 2528 . . . . . 6  |-  F/_ n { f  |  E. n  e.  D  (
f  Fn  n  /\  ph 
/\  ps ) }
111, 10nfcxfr 2521 . . . . 5  |-  F/_ n B
1211nfcri 2518 . . . 4  |-  F/ n  f  e.  B
131219.37 1883 . . 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 201 . 2  |-  E. n
( f  e.  B  ->  ( n  e.  D  /\  dom  f  =  n ) )
15 nfv 1626 . . . 4  |-  F/ n (/) 
e.  dom  f
1612, 15nfim 1822 . . 3  |-  F/ n
( f  e.  B  -> 
(/)  e.  dom  f )
17 bnj900.3 . . . . . 6  |-  D  =  ( om  \  { (/)
} )
1817bnj529 28448 . . . . 5  |-  ( n  e.  D  ->  (/)  e.  n
)
19 eleq2 2449 . . . . . 6  |-  ( dom  f  =  n  -> 
( (/)  e.  dom  f  <->  (/)  e.  n ) )
2019biimparc 474 . . . . 5  |-  ( (
(/)  e.  n  /\  dom  f  =  n
)  ->  (/)  e.  dom  f )
2118, 20sylan 458 . . . 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 1811 . 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 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2374   E.wrex 2651    \ cdif 3261   (/)c0 3572   {csn 3758   omcom 4786   dom cdm 4819    Fn wfn 5390
This theorem is referenced by:  bnj906  28640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-tr 4245  df-eprel 4436  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-fn 5398
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