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Theorem bnj1234 28721
Description: Technical lemma for bnj60 28770. 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 5475 . . . . 5  |-  ( f  =  g  ->  (
f  Fn  d  <->  g  Fn  d ) )
2 fveq1 5668 . . . . . . 7  |-  ( f  =  g  ->  (
f `  x )  =  ( g `  x ) )
3 reseq1 5081 . . . . . . . . . 10  |-  ( f  =  g  ->  (
f  |`  pred ( x ,  A ,  R ) )  =  ( g  |`  pred ( x ,  A ,  R ) ) )
43opeq2d 3934 . . . . . . . . 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 2445 . . . . . . . 8  |-  ( f  =  g  ->  Y  =  Z )
87fveq2d 5673 . . . . . . 7  |-  ( f  =  g  ->  ( G `  Y )  =  ( G `  Z ) )
92, 8eqeq12d 2402 . . . . . 6  |-  ( f  =  g  ->  (
( f `  x
)  =  ( G `
 Y )  <->  ( g `  x )  =  ( G `  Z ) ) )
109ralbidv 2670 . . . . 5  |-  ( f  =  g  ->  ( A. x  e.  d 
( f `  x
)  =  ( G `
 Y )  <->  A. x  e.  d  ( g `  x )  =  ( G `  Z ) ) )
111, 10anbi12d 692 . . . 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 2671 . . 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 2507 . 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 2418 1  |-  C  =  D
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649   {cab 2374   A.wral 2650   E.wrex 2651   <.cop 3761    |` cres 4821    Fn wfn 5390   ` cfv 5395    predc-bnj14 28391
This theorem is referenced by:  bnj1245  28722  bnj1256  28723  bnj1259  28724  bnj1296  28729  bnj1311  28732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-res 4831  df-iota 5359  df-fun 5397  df-fn 5398  df-fv 5403
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