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Theorem bnj66 29208
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
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 5349 . . . . . . 7  |-  ( g  =  f  ->  (
g  Fn  d  <->  f  Fn  d ) )
3 fveq1 5540 . . . . . . . . 9  |-  ( g  =  f  ->  (
g `  x )  =  ( f `  x ) )
4 reseq1 4965 . . . . . . . . . . . 12  |-  ( g  =  f  ->  (
g  |`  pred ( x ,  A ,  R ) )  =  ( f  |`  pred ( x ,  A ,  R ) ) )
54opeq2d 3819 . . . . . . . . . . 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 2346 . . . . . . . . . 10  |-  ( g  =  f  ->  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.  =  Y )
87fveq2d 5545 . . . . . . . . 9  |-  ( g  =  f  ->  ( G `  <. x ,  ( g  |`  pred (
x ,  A ,  R ) ) >.
)  =  ( G `
 Y ) )
93, 8eqeq12d 2310 . . . . . . . 8  |-  ( g  =  f  ->  (
( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )  <->  ( f `  x )  =  ( G `  Y ) ) )
109ralbidv 2576 . . . . . . 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 691 . . . . . 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 2577 . . . . 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 2415 . . . 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 2319 . . 3  |-  C  =  { g  |  E. d  e.  B  (
g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >. ) ) }
1514bnj1436 29188 . 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 29154 . 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 5358 . . 3  |-  ( g  Fn  d  ->  Rel  g )
1817rexlimivw 2676 . 2  |-  ( E. d  e.  B  g  Fn  d  ->  Rel  g )
1915, 16, 183syl 18 1  |-  ( g  e.  C  ->  Rel  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   Rel wrel 4710    Fn wfn 5266   ` cfv 5271    predc-bnj14 29029
This theorem is referenced by:  bnj1321  29373
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
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