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Theorem bnj1497 29941
Description: Technical lemma for bnj60 29943. 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 29705 . . . . 5  |-  ( g  e.  C  ->  A. f 
g  e.  C )
32nfi 1682 . . . 4  |-  F/ f  g  e.  C
4 nfv 1769 . . . 4  |-  F/ f Fun  g
53, 4nfim 2023 . . 3  |-  F/ f ( g  e.  C  ->  Fun  g )
6 eleq1 2537 . . . 4  |-  ( f  =  g  ->  (
f  e.  C  <->  g  e.  C ) )
7 funeq 5608 . . . 4  |-  ( f  =  g  ->  ( Fun  f  <->  Fun  g ) )
86, 7imbi12d 327 . . 3  |-  ( f  =  g  ->  (
( f  e.  C  ->  Fun  f )  <->  ( g  e.  C  ->  Fun  g
) ) )
91bnj1436 29723 . . . . . 6  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
109bnj1299 29702 . . . . 5  |-  ( f  e.  C  ->  E. d  e.  B  f  Fn  d )
11 fnfun 5683 . . . . 5  |-  ( f  Fn  d  ->  Fun  f )
1210, 11bnj31 29597 . . . 4  |-  ( f  e.  C  ->  E. d  e.  B  Fun  f )
1312bnj1265 29696 . . 3  |-  ( f  e.  C  ->  Fun  f )
145, 8, 13chvar 2119 . 2  |-  ( g  e.  C  ->  Fun  g )
1514rgen 2766 1  |-  A. g  e.  C  Fun  g
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   {cab 2457   A.wral 2756   E.wrex 2757    C_ wss 3390   <.cop 3965    |` cres 4841   Fun wfun 5583    Fn wfn 5584   ` cfv 5589    predc-bnj14 29565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-in 3397  df-ss 3404  df-br 4396  df-opab 4455  df-rel 4846  df-cnv 4847  df-co 4848  df-fun 5591  df-fn 5592
This theorem is referenced by:  bnj60  29943
  Copyright terms: Public domain W3C validator