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Theorem bnj1234 29043
Description: Technical lemma for bnj60 29092. 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
bnj1234.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1234.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1234.4  |-  Z  = 
<. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.
bnj1234.5  |-  D  =  { g  |  E. d  e.  B  (
g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  Z ) ) }
Assertion
Ref Expression
bnj1234  |-  C  =  D
Distinct variable groups:    B, f,
g    f, G, g    g, Y    f, Z    f, d,
g    x, f, g
Allowed substitution hints:    A( x, f, g, d)    B( x, d)    C( x, f, g, d)    D( x, f, g, d)    R( x, f, g, d)    G( x, d)    Y( x, f, d)    Z( x, g, d)

Proof of Theorem bnj1234
StepHypRef Expression
1 fneq1 5333 . . . . 5  |-  ( f  =  g  ->  (
f  Fn  d  <->  g  Fn  d ) )
2 fveq1 5524 . . . . . . 7  |-  ( f  =  g  ->  (
f `  x )  =  ( g `  x ) )
3 reseq1 4949 . . . . . . . . . 10  |-  ( f  =  g  ->  (
f  |`  pred ( x ,  A ,  R ) )  =  ( g  |`  pred ( x ,  A ,  R ) ) )
43opeq2d 3803 . . . . . . . . 9  |-  ( f  =  g  ->  <. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.  =  <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )
5 bnj1234.2 . . . . . . . . 9  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
6 bnj1234.4 . . . . . . . . 9  |-  Z  = 
<. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.
74, 5, 63eqtr4g 2340 . . . . . . . 8  |-  ( f  =  g  ->  Y  =  Z )
87fveq2d 5529 . . . . . . 7  |-  ( f  =  g  ->  ( G `  Y )  =  ( G `  Z ) )
92, 8eqeq12d 2297 . . . . . 6  |-  ( f  =  g  ->  (
( f `  x
)  =  ( G `
 Y )  <->  ( g `  x )  =  ( G `  Z ) ) )
109ralbidv 2563 . . . . 5  |-  ( f  =  g  ->  ( A. x  e.  d 
( f `  x
)  =  ( G `
 Y )  <->  A. x  e.  d  ( g `  x )  =  ( G `  Z ) ) )
111, 10anbi12d 691 . . . 4  |-  ( f  =  g  ->  (
( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )  <-> 
( g  Fn  d  /\  A. x  e.  d  ( g `  x
)  =  ( G `
 Z ) ) ) )
1211rexbidv 2564 . . 3  |-  ( f  =  g  ->  ( E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x
)  =  ( G `
 Y ) )  <->  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x
)  =  ( G `
 Z ) ) ) )
1312cbvabv 2402 . 2  |-  { f  |  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }  =  {
g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  Z ) ) }
14 bnj1234.3 . 2  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
15 bnj1234.5 . 2  |-  D  =  { g  |  E. d  e.  B  (
g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  Z ) ) }
1613, 14, 153eqtr4i 2313 1  |-  C  =  D
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623   {cab 2269   A.wral 2543   E.wrex 2544   <.cop 3643    |` cres 4691    Fn wfn 5250   ` cfv 5255    predc-bnj14 28713
This theorem is referenced by:  bnj1245  29044  bnj1256  29045  bnj1259  29046  bnj1296  29051  bnj1311  29054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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