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Theorem bnj1296 28803
Description: Technical lemma for bnj60 28844. 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
bnj1296.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1296.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1296.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1296.4  |-  D  =  ( dom  g  i^i 
dom  h )
bnj1296.5  |-  E  =  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
bnj1296.6  |-  ( ph  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
bnj1296.7  |-  ( ps  <->  (
ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )
bnj1296.18  |-  ( ps 
->  ( g  |`  pred (
x ,  A ,  R ) )  =  ( h  |`  pred (
x ,  A ,  R ) ) )
bnj1296.9  |-  Z  = 
<. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.
bnj1296.10  |-  K  =  { g  |  E. d  e.  B  (
g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  Z ) ) }
bnj1296.11  |-  W  = 
<. x ,  ( h  |`  pred ( x ,  A ,  R ) ) >.
bnj1296.12  |-  L  =  { h  |  E. d  e.  B  (
h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  W ) ) }
Assertion
Ref Expression
bnj1296  |-  ( ps 
->  ( g `  x
)  =  ( h `
 x ) )
Distinct variable groups:    B, f,
g    B, h, f    x, D    G, d, f, g   
h, G, d    W, d, f    g, Y    h, Y    Z, d, f    x, d, f, g    x, h
Allowed substitution hints:    ph( x, y, f, g, h, d)    ps( x, y, f, g, h, d)    A( x, y, f, g, h, d)    B( x, y, d)    C( x, y, f, g, h, d)    D( y, f, g, h, d)    R( x, y, f, g, h, d)    E( x, y, f, g, h, d)    G( x, y)    K( x, y, f, g, h, d)    L( x, y, f, g, h, d)    W( x, y, g, h)    Y( x, y, f, d)    Z( x, y, g, h)

Proof of Theorem bnj1296
StepHypRef Expression
1 bnj1296.18 . . . . 5  |-  ( ps 
->  ( g  |`  pred (
x ,  A ,  R ) )  =  ( h  |`  pred (
x ,  A ,  R ) ) )
21opeq2d 3905 . . . 4  |-  ( ps 
->  <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >.  =  <. x ,  ( h  |`  pred ( x ,  A ,  R ) ) >.
)
3 bnj1296.9 . . . 4  |-  Z  = 
<. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.
4 bnj1296.11 . . . 4  |-  W  = 
<. x ,  ( h  |`  pred ( x ,  A ,  R ) ) >.
52, 3, 43eqtr4g 2423 . . 3  |-  ( ps 
->  Z  =  W
)
65fveq2d 5636 . 2  |-  ( ps 
->  ( G `  Z
)  =  ( G `
 W ) )
7 bnj1296.7 . . . 4  |-  ( ps  <->  (
ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )
8 bnj1296.6 . . . . 5  |-  ( ph  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
9 bnj1296.10 . . . . . . . . . . 11  |-  K  =  { g  |  E. d  e.  B  (
g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  Z ) ) }
109bnj1436 28624 . . . . . . . . . 10  |-  ( g  e.  K  ->  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  Z ) ) )
11 fndm 5448 . . . . . . . . . . 11  |-  ( g  Fn  d  ->  dom  g  =  d )
1211anim1i 551 . . . . . . . . . 10  |-  ( ( g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  Z ) )  -> 
( dom  g  =  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  Z ) ) )
1310, 12bnj31 28497 . . . . . . . . 9  |-  ( g  e.  K  ->  E. d  e.  B  ( dom  g  =  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  Z ) ) )
14 raleq 2821 . . . . . . . . . . 11  |-  ( dom  g  =  d  -> 
( A. x  e. 
dom  g ( g `
 x )  =  ( G `  Z
)  <->  A. x  e.  d  ( g `  x
)  =  ( G `
 Z ) ) )
1514pm5.32i 618 . . . . . . . . . 10  |-  ( ( dom  g  =  d  /\  A. x  e. 
dom  g ( g `
 x )  =  ( G `  Z
) )  <->  ( dom  g  =  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  Z ) ) )
1615rexbii 2653 . . . . . . . . 9  |-  ( E. d  e.  B  ( dom  g  =  d  /\  A. x  e. 
dom  g ( g `
 x )  =  ( G `  Z
) )  <->  E. d  e.  B  ( dom  g  =  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  Z ) ) )
1713, 16sylibr 203 . . . . . . . 8  |-  ( g  e.  K  ->  E. d  e.  B  ( dom  g  =  d  /\  A. x  e.  dom  g
( g `  x
)  =  ( G `
 Z ) ) )
18 simpr 447 . . . . . . . 8  |-  ( ( dom  g  =  d  /\  A. x  e. 
dom  g ( g `
 x )  =  ( G `  Z
) )  ->  A. x  e.  dom  g ( g `
 x )  =  ( G `  Z
) )
1917, 18bnj31 28497 . . . . . . 7  |-  ( g  e.  K  ->  E. d  e.  B  A. x  e.  dom  g ( g `
 x )  =  ( G `  Z
) )
2019bnj1265 28597 . . . . . 6  |-  ( g  e.  K  ->  A. x  e.  dom  g ( g `
 x )  =  ( G `  Z
) )
21 bnj1296.2 . . . . . . 7  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
22 bnj1296.3 . . . . . . 7  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
2321, 22, 3, 9bnj1234 28795 . . . . . 6  |-  C  =  K
2420, 23eleq2s 2458 . . . . 5  |-  ( g  e.  C  ->  A. x  e.  dom  g ( g `
 x )  =  ( G `  Z
) )
258, 24bnj770 28545 . . . 4  |-  ( ph  ->  A. x  e.  dom  g ( g `  x )  =  ( G `  Z ) )
267, 25bnj835 28541 . . 3  |-  ( ps 
->  A. x  e.  dom  g ( g `  x )  =  ( G `  Z ) )
27 bnj1296.4 . . . . 5  |-  D  =  ( dom  g  i^i 
dom  h )
2827bnj1292 28600 . . . 4  |-  D  C_  dom  g
29 bnj1296.5 . . . . 5  |-  E  =  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
3029, 7bnj1212 28584 . . . 4  |-  ( ps 
->  x  e.  D
)
3128, 30bnj1213 28583 . . 3  |-  ( ps 
->  x  e.  dom  g )
3226, 31bnj1294 28602 . 2  |-  ( ps 
->  ( g `  x
)  =  ( G `
 Z ) )
33 bnj1296.12 . . . . . . . . . . 11  |-  L  =  { h  |  E. d  e.  B  (
h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  W ) ) }
3433bnj1436 28624 . . . . . . . . . 10  |-  ( h  e.  L  ->  E. d  e.  B  ( h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  W ) ) )
35 fndm 5448 . . . . . . . . . . 11  |-  ( h  Fn  d  ->  dom  h  =  d )
3635anim1i 551 . . . . . . . . . 10  |-  ( ( h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  W ) )  -> 
( dom  h  =  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  W ) ) )
3734, 36bnj31 28497 . . . . . . . . 9  |-  ( h  e.  L  ->  E. d  e.  B  ( dom  h  =  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  W ) ) )
38 raleq 2821 . . . . . . . . . . 11  |-  ( dom  h  =  d  -> 
( A. x  e. 
dom  h ( h `
 x )  =  ( G `  W
)  <->  A. x  e.  d  ( h `  x
)  =  ( G `
 W ) ) )
3938pm5.32i 618 . . . . . . . . . 10  |-  ( ( dom  h  =  d  /\  A. x  e. 
dom  h ( h `
 x )  =  ( G `  W
) )  <->  ( dom  h  =  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  W ) ) )
4039rexbii 2653 . . . . . . . . 9  |-  ( E. d  e.  B  ( dom  h  =  d  /\  A. x  e. 
dom  h ( h `
 x )  =  ( G `  W
) )  <->  E. d  e.  B  ( dom  h  =  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  W ) ) )
4137, 40sylibr 203 . . . . . . . 8  |-  ( h  e.  L  ->  E. d  e.  B  ( dom  h  =  d  /\  A. x  e.  dom  h
( h `  x
)  =  ( G `
 W ) ) )
42 simpr 447 . . . . . . . 8  |-  ( ( dom  h  =  d  /\  A. x  e. 
dom  h ( h `
 x )  =  ( G `  W
) )  ->  A. x  e.  dom  h ( h `
 x )  =  ( G `  W
) )
4341, 42bnj31 28497 . . . . . . 7  |-  ( h  e.  L  ->  E. d  e.  B  A. x  e.  dom  h ( h `
 x )  =  ( G `  W
) )
4443bnj1265 28597 . . . . . 6  |-  ( h  e.  L  ->  A. x  e.  dom  h ( h `
 x )  =  ( G `  W
) )
4521, 22, 4, 33bnj1234 28795 . . . . . 6  |-  C  =  L
4644, 45eleq2s 2458 . . . . 5  |-  ( h  e.  C  ->  A. x  e.  dom  h ( h `
 x )  =  ( G `  W
) )
478, 46bnj771 28546 . . . 4  |-  ( ph  ->  A. x  e.  dom  h ( h `  x )  =  ( G `  W ) )
487, 47bnj835 28541 . . 3  |-  ( ps 
->  A. x  e.  dom  h ( h `  x )  =  ( G `  W ) )
4927bnj1293 28601 . . . 4  |-  D  C_  dom  h
5049, 30bnj1213 28583 . . 3  |-  ( ps 
->  x  e.  dom  h )
5148, 50bnj1294 28602 . 2  |-  ( ps 
->  ( h `  x
)  =  ( G `
 W ) )
526, 32, 513eqtr4d 2408 1  |-  ( ps 
->  ( g `  x
)  =  ( h `
 x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   {cab 2352    =/= wne 2529   A.wral 2628   E.wrex 2629   {crab 2632    i^i cin 3237    C_ wss 3238   <.cop 3732   class class class wbr 4125   dom cdm 4792    |` cres 4794    Fn wfn 5353   ` cfv 5358    /\ w-bnj17 28463    predc-bnj14 28465    FrSe w-bnj15 28469
This theorem is referenced by:  bnj1311  28806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-res 4804  df-iota 5322  df-fun 5360  df-fn 5361  df-fv 5366  df-bnj17 28464
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