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Theorem bnj1259 34215
Description: Technical lemma for bnj60 34261. 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
bnj1259.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1259.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1259.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1259.4  |-  D  =  ( dom  g  i^i 
dom  h )
bnj1259.5  |-  E  =  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
bnj1259.6  |-  ( ph  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
bnj1259.7  |-  ( ps  <->  (
ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )
Assertion
Ref Expression
bnj1259  |-  ( ph  ->  E. d  e.  B  h  Fn  d )
Distinct variable groups:    A, f    B, f, h    f, G, h    R, f    h, Y   
f, d, h    x, f, h
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, g, 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, g, d)    Y( x, y, f, g, d)

Proof of Theorem bnj1259
StepHypRef Expression
1 bnj1259.6 . 2  |-  ( ph  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
2 abid 2444 . . . 4  |-  ( h  e.  { h  |  E. d  e.  B  ( h  Fn  d  /\  A. x  e.  d  ( h `  x
)  =  ( G `
 <. x ,  ( h  |`  pred ( x ,  A ,  R
) ) >. )
) }  <->  E. d  e.  B  ( h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  <. x ,  ( h  |`  pred ( x ,  A ,  R ) ) >.
) ) )
32bnj1238 34008 . . 3  |-  ( h  e.  { h  |  E. d  e.  B  ( h  Fn  d  /\  A. x  e.  d  ( h `  x
)  =  ( G `
 <. x ,  ( h  |`  pred ( x ,  A ,  R
) ) >. )
) }  ->  E. d  e.  B  h  Fn  d )
4 bnj1259.2 . . . 4  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
5 bnj1259.3 . . . 4  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
6 eqid 2457 . . . 4  |-  <. x ,  ( h  |`  pred ( x ,  A ,  R ) ) >.  =  <. x ,  ( h  |`  pred ( x ,  A ,  R
) ) >.
7 eqid 2457 . . . 4  |-  { h  |  E. d  e.  B  ( h  Fn  d  /\  A. x  e.  d  ( h `  x
)  =  ( G `
 <. x ,  ( h  |`  pred ( x ,  A ,  R
) ) >. )
) }  =  {
h  |  E. d  e.  B  ( h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  <. x ,  ( h  |`  pred ( x ,  A ,  R ) ) >.
) ) }
84, 5, 6, 7bnj1234 34212 . . 3  |-  C  =  { h  |  E. d  e.  B  (
h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  <. x ,  ( h  |`  pred ( x ,  A ,  R ) ) >. ) ) }
93, 8eleq2s 2565 . 2  |-  ( h  e.  C  ->  E. d  e.  B  h  Fn  d )
101, 9bnj771 33965 1  |-  ( ph  ->  E. d  e.  B  h  Fn  d )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   {cab 2442    =/= wne 2652   A.wral 2807   E.wrex 2808   {crab 2811    i^i cin 3470    C_ wss 3471   <.cop 4038   class class class wbr 4456   dom cdm 5008    |` cres 5010    Fn wfn 5589   ` cfv 5594    /\ w-bnj17 33881    predc-bnj14 33883    FrSe w-bnj15 33887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-res 5020  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-bnj17 33882
This theorem is referenced by:  bnj1253  34216  bnj1286  34218  bnj1280  34219
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