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Theorem bnj1497 29090
Description: Technical lemma for bnj60 29092. 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
bnj1497.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1497.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1497.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
Assertion
Ref Expression
bnj1497  |-  A. g  e.  C  Fun  g
Distinct variable groups:    C, g    f, d    f, g
Allowed substitution hints:    A( x, f, g, d)    B( x, f, g, d)    C( x, f, d)    R( x, f, g, d)    G( x, f, g, d)    Y( x, f, g, d)

Proof of Theorem bnj1497
StepHypRef Expression
1 bnj1497.3 . . . . . 6  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
21bnj1317 28854 . . . . 5  |-  ( g  e.  C  ->  A. f 
g  e.  C )
32nfi 1538 . . . 4  |-  F/ f  g  e.  C
4 nfv 1605 . . . 4  |-  F/ f Fun  g
53, 4nfim 1769 . . 3  |-  F/ f ( g  e.  C  ->  Fun  g )
6 eleq1 2343 . . . 4  |-  ( f  =  g  ->  (
f  e.  C  <->  g  e.  C ) )
7 funeq 5274 . . . 4  |-  ( f  =  g  ->  ( Fun  f  <->  Fun  g ) )
86, 7imbi12d 311 . . 3  |-  ( f  =  g  ->  (
( f  e.  C  ->  Fun  f )  <->  ( g  e.  C  ->  Fun  g
) ) )
91bnj1436 28872 . . . . . 6  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
109bnj1299 28851 . . . . 5  |-  ( f  e.  C  ->  E. d  e.  B  f  Fn  d )
11 fnfun 5341 . . . . 5  |-  ( f  Fn  d  ->  Fun  f )
1210, 11bnj31 28745 . . . 4  |-  ( f  e.  C  ->  E. d  e.  B  Fun  f )
1312bnj1265 28845 . . 3  |-  ( f  e.  C  ->  Fun  f )
145, 8, 13chvar 1926 . 2  |-  ( g  e.  C  ->  Fun  g )
1514rgen 2608 1  |-  A. g  e.  C  Fun  g
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544    C_ wss 3152   <.cop 3643    |` cres 4691   Fun wfun 5249    Fn wfn 5250   ` cfv 5255    predc-bnj14 28713
This theorem is referenced by:  bnj60  29092
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-in 3159  df-ss 3166  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-fun 5257  df-fn 5258
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