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Theorem bnj66 32872
Description: Technical lemma for bnj60 33072. 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 5660 . . . . . . 7  |-  ( g  =  f  ->  (
g  Fn  d  <->  f  Fn  d ) )
3 fveq1 5856 . . . . . . . . 9  |-  ( g  =  f  ->  (
g `  x )  =  ( f `  x ) )
4 reseq1 5258 . . . . . . . . . . . 12  |-  ( g  =  f  ->  (
g  |`  pred ( x ,  A ,  R ) )  =  ( f  |`  pred ( x ,  A ,  R ) ) )
54opeq2d 4213 . . . . . . . . . . 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 2519 . . . . . . . . . 10  |-  ( g  =  f  ->  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.  =  Y )
87fveq2d 5861 . . . . . . . . 9  |-  ( g  =  f  ->  ( G `  <. x ,  ( g  |`  pred (
x ,  A ,  R ) ) >.
)  =  ( G `
 Y ) )
93, 8eqeq12d 2482 . . . . . . . 8  |-  ( g  =  f  ->  (
( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )  <->  ( f `  x )  =  ( G `  Y ) ) )
109ralbidv 2896 . . . . . . 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 2966 . . . . 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 2603 . . . 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 2492 . . 3  |-  C  =  { g  |  E. d  e.  B  (
g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >. ) ) }
1514bnj1436 32852 . 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 32818 . 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 5670 . . 3  |-  ( g  Fn  d  ->  Rel  g )
1817rexlimivw 2945 . 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 1374    e. wcel 1762   {cab 2445   A.wral 2807   E.wrex 2808    C_ wss 3469   <.cop 4026    |` cres 4994   Rel wrel 4997    Fn wfn 5574   ` cfv 5579    predc-bnj14 32695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-res 5004  df-iota 5542  df-fun 5581  df-fn 5582  df-fv 5587
This theorem is referenced by:  bnj1321  33037
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