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Theorem bnj66 31949
Description: Technical lemma for bnj60 32149. 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
bnj66.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj66.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj66.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
Assertion
Ref Expression
bnj66  |-  ( g  e.  C  ->  Rel  g )
Distinct variable groups:    A, f    B, f, g    f, G, g    R, f    g, Y   
f, d, g    x, f, g
Allowed substitution hints:    A( x, g, d)    B( x, d)    C( x, f, g, d)    R( x, g, d)    G( x, d)    Y( x, f, d)

Proof of Theorem bnj66
StepHypRef Expression
1 bnj66.3 . . . 4  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
2 fneq1 5520 . . . . . . 7  |-  ( g  =  f  ->  (
g  Fn  d  <->  f  Fn  d ) )
3 fveq1 5711 . . . . . . . . 9  |-  ( g  =  f  ->  (
g `  x )  =  ( f `  x ) )
4 reseq1 5125 . . . . . . . . . . . 12  |-  ( g  =  f  ->  (
g  |`  pred ( x ,  A ,  R ) )  =  ( f  |`  pred ( x ,  A ,  R ) ) )
54opeq2d 4087 . . . . . . . . . . 11  |-  ( g  =  f  ->  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.  =  <. x ,  ( f  |`  pred ( x ,  A ,  R
) ) >. )
6 bnj66.2 . . . . . . . . . . 11  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
75, 6syl6eqr 2493 . . . . . . . . . 10  |-  ( g  =  f  ->  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.  =  Y )
87fveq2d 5716 . . . . . . . . 9  |-  ( g  =  f  ->  ( G `  <. x ,  ( g  |`  pred (
x ,  A ,  R ) ) >.
)  =  ( G `
 Y ) )
93, 8eqeq12d 2457 . . . . . . . 8  |-  ( g  =  f  ->  (
( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )  <->  ( f `  x )  =  ( G `  Y ) ) )
109ralbidv 2756 . . . . . . 7  |-  ( g  =  f  ->  ( A. x  e.  d 
( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )  <->  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
112, 10anbi12d 710 . . . . . 6  |-  ( g  =  f  ->  (
( g  Fn  d  /\  A. x  e.  d  ( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )
)  <->  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) ) )
1211rexbidv 2757 . . . . 5  |-  ( g  =  f  ->  ( E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )
)  <->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) ) ) )
1312cbvabv 2570 . . . 4  |-  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.
) ) }  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
141, 13eqtr4i 2466 . . 3  |-  C  =  { g  |  E. d  e.  B  (
g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >. ) ) }
1514bnj1436 31929 . 2  |-  ( g  e.  C  ->  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.
) ) )
16 bnj1239 31895 . 2  |-  ( 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 )
17 fnrel 5530 . . 3  |-  ( g  Fn  d  ->  Rel  g )
1817rexlimivw 2858 . 2  |-  ( E. d  e.  B  g  Fn  d  ->  Rel  g )
1915, 16, 183syl 20 1  |-  ( g  e.  C  ->  Rel  g )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2736   E.wrex 2737    C_ wss 3349   <.cop 3904    |` cres 4863   Rel wrel 4866    Fn wfn 5434   ` cfv 5439    predc-bnj14 31772
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-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-res 4873  df-iota 5402  df-fun 5441  df-fn 5442  df-fv 5447
This theorem is referenced by:  bnj1321  32114
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