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Theorem bnj1497 32056
Description: Technical lemma for bnj60 32058. 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 31820 . . . . 5  |-  ( g  e.  C  ->  A. f 
g  e.  C )
32nfi 1596 . . . 4  |-  F/ f  g  e.  C
4 nfv 1673 . . . 4  |-  F/ f Fun  g
53, 4nfim 1853 . . 3  |-  F/ f ( g  e.  C  ->  Fun  g )
6 eleq1 2503 . . . 4  |-  ( f  =  g  ->  (
f  e.  C  <->  g  e.  C ) )
7 funeq 5442 . . . 4  |-  ( f  =  g  ->  ( Fun  f  <->  Fun  g ) )
86, 7imbi12d 320 . . 3  |-  ( f  =  g  ->  (
( f  e.  C  ->  Fun  f )  <->  ( g  e.  C  ->  Fun  g
) ) )
91bnj1436 31838 . . . . . 6  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
109bnj1299 31817 . . . . 5  |-  ( f  e.  C  ->  E. d  e.  B  f  Fn  d )
11 fnfun 5513 . . . . 5  |-  ( f  Fn  d  ->  Fun  f )
1210, 11bnj31 31713 . . . 4  |-  ( f  e.  C  ->  E. d  e.  B  Fun  f )
1312bnj1265 31811 . . 3  |-  ( f  e.  C  ->  Fun  f )
145, 8, 13chvar 1957 . 2  |-  ( g  e.  C  ->  Fun  g )
1514rgen 2786 1  |-  A. g  e.  C  Fun  g
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2720   E.wrex 2721    C_ wss 3333   <.cop 3888    |` cres 4847   Fun wfun 5417    Fn wfn 5418   ` cfv 5423    predc-bnj14 31681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-rex 2726  df-in 3340  df-ss 3347  df-br 4298  df-opab 4356  df-rel 4852  df-cnv 4853  df-co 4854  df-fun 5425  df-fn 5426
This theorem is referenced by:  bnj60  32058
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