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Theorem bnj1256 29361
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
bnj1256.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1256.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1256.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1256.4  |-  D  =  ( dom  g  i^i 
dom  h )
bnj1256.5  |-  E  =  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
bnj1256.6  |-  ( ph  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
bnj1256.7  |-  ( ps  <->  (
ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )
Assertion
Ref Expression
bnj1256  |-  ( ph  ->  E. d  e.  B  g  Fn  d )
Distinct variable groups:    A, f    B, f, g    f, G, g    R, f    g, Y   
f, d, g    x, f, g
Allowed substitution hints:    ph( x, y, f, g, h, d)    ps( x, y, f, g, h, d)    A( x, y, g, h, d)    B( x, y, h, d)    C( x, y, f, g, h, d)    D( x, y, f, g, h, d)    R( x, y, g, h, d)    E( x, y, f, g, h, d)    G( x, y, h, d)    Y( x, y, f, h, d)

Proof of Theorem bnj1256
StepHypRef Expression
1 bnj1256.6 . 2  |-  ( ph  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
2 abid 2284 . . . 4  |-  ( g  e.  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )
) }  <->  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.
) ) )
32bnj1238 29155 . . 3  |-  ( g  e.  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )
) }  ->  E. d  e.  B  g  Fn  d )
4 bnj1256.2 . . . 4  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
5 bnj1256.3 . . . 4  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
6 eqid 2296 . . . 4  |-  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.  =  <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >.
7 eqid 2296 . . . 4  |-  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.
) ) }  =  { g  |  E. d  e.  B  (
g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >. ) ) }
84, 5, 6, 7bnj1234 29359 . . 3  |-  C  =  { g  |  E. d  e.  B  (
g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >. ) ) }
93, 8eleq2s 2388 . 2  |-  ( g  e.  C  ->  E. d  e.  B  g  Fn  d )
101, 9bnj770 29109 1  |-  ( ph  ->  E. d  e.  B  g  Fn  d )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    i^i cin 3164    C_ wss 3165   <.cop 3656   class class class wbr 4039   dom cdm 4705    |` cres 4707    Fn wfn 5266   ` cfv 5271    /\ w-bnj17 29027    predc-bnj14 29029    FrSe w-bnj15 29033
This theorem is referenced by:  bnj1253  29363  bnj1286  29365  bnj1280  29366
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-3an 936  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-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-bnj17 29028
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