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Theorem bnj1497 29406
Description: Technical lemma for bnj60 29408. 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 29170 . . . . 5  |-  ( g  e.  C  ->  A. f 
g  e.  C )
32nfi 1541 . . . 4  |-  F/ f  g  e.  C
4 nfv 1609 . . . 4  |-  F/ f Fun  g
53, 4nfim 1781 . . 3  |-  F/ f ( g  e.  C  ->  Fun  g )
6 eleq1 2356 . . . 4  |-  ( f  =  g  ->  (
f  e.  C  <->  g  e.  C ) )
7 funeq 5290 . . . 4  |-  ( f  =  g  ->  ( Fun  f  <->  Fun  g ) )
86, 7imbi12d 311 . . 3  |-  ( f  =  g  ->  (
( f  e.  C  ->  Fun  f )  <->  ( g  e.  C  ->  Fun  g
) ) )
91bnj1436 29188 . . . . . 6  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
109bnj1299 29167 . . . . 5  |-  ( f  e.  C  ->  E. d  e.  B  f  Fn  d )
11 fnfun 5357 . . . . 5  |-  ( f  Fn  d  ->  Fun  f )
1210, 11bnj31 29061 . . . 4  |-  ( f  e.  C  ->  E. d  e.  B  Fun  f )
1312bnj1265 29161 . . 3  |-  ( f  e.  C  ->  Fun  f )
145, 8, 13chvar 1939 . 2  |-  ( g  e.  C  ->  Fun  g )
1514rgen 2621 1  |-  A. g  e.  C  Fun  g
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557    C_ wss 3165   <.cop 3656    |` cres 4707   Fun wfun 5265    Fn wfn 5266   ` cfv 5271    predc-bnj14 29029
This theorem is referenced by:  bnj60  29408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-in 3172  df-ss 3179  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-fun 5273  df-fn 5274
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