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Theorem bnj1280 29781
Description: Technical lemma for bnj60 29823. 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
bnj1280.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1280.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1280.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1280.4  |-  D  =  ( dom  g  i^i 
dom  h )
bnj1280.5  |-  E  =  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
bnj1280.6  |-  ( ph  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
bnj1280.7  |-  ( ps  <->  (
ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )
bnj1280.17  |-  ( ps 
->  (  pred ( x ,  A ,  R
)  i^i  E )  =  (/) )
Assertion
Ref Expression
bnj1280  |-  ( ps 
->  ( g  |`  pred (
x ,  A ,  R ) )  =  ( h  |`  pred (
x ,  A ,  R ) ) )
Distinct variable groups:    A, d,
f    B, f, g    B, h, f    D, d, x   
f, G, g    h, G    R, d, f    g, Y    h, Y    g, d    x, f, g    h, d, x
Allowed substitution hints:    ph( x, y, f, g, h, d)    ps( x, y, f, g, h, d)    A( x, y, g, h)    B( x, y, d)    C( x, y, f, g, h, d)    D( y, f, g, h)    R( x, y, g, h)    E( x, y, f, g, h, d)    G( x, y, d)    Y( x, y, f, d)

Proof of Theorem bnj1280
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 bnj1280.1 . . . . . . . 8  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
2 bnj1280.2 . . . . . . . 8  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
3 bnj1280.3 . . . . . . . 8  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
4 bnj1280.4 . . . . . . . 8  |-  D  =  ( dom  g  i^i 
dom  h )
5 bnj1280.5 . . . . . . . 8  |-  E  =  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
6 bnj1280.6 . . . . . . . 8  |-  ( ph  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
7 bnj1280.7 . . . . . . . 8  |-  ( ps  <->  (
ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )
81, 2, 3, 4, 5, 6, 7bnj1286 29780 . . . . . . 7  |-  ( ps 
->  pred ( x ,  A ,  R ) 
C_  D )
98sseld 3406 . . . . . 6  |-  ( ps 
->  ( z  e.  pred ( x ,  A ,  R )  ->  z  e.  D ) )
10 bnj1280.17 . . . . . . . . 9  |-  ( ps 
->  (  pred ( x ,  A ,  R
)  i^i  E )  =  (/) )
11 disj1 3780 . . . . . . . . 9  |-  ( ( 
pred ( x ,  A ,  R )  i^i  E )  =  (/) 
<-> 
A. z ( z  e.  pred ( x ,  A ,  R )  ->  -.  z  e.  E ) )
1210, 11sylib 199 . . . . . . . 8  |-  ( ps 
->  A. z ( z  e.  pred ( x ,  A ,  R )  ->  -.  z  e.  E ) )
131219.21bi 1924 . . . . . . 7  |-  ( ps 
->  ( z  e.  pred ( x ,  A ,  R )  ->  -.  z  e.  E )
)
14 fveq2 5825 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
g `  x )  =  ( g `  z ) )
15 fveq2 5825 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
h `  x )  =  ( h `  z ) )
1614, 15neeq12d 2662 . . . . . . . . . 10  |-  ( x  =  z  ->  (
( g `  x
)  =/=  ( h `
 x )  <->  ( g `  z )  =/=  (
h `  z )
) )
1716, 5elrab2 3173 . . . . . . . . 9  |-  ( z  e.  E  <->  ( z  e.  D  /\  (
g `  z )  =/=  ( h `  z
) ) )
1817notbii 297 . . . . . . . 8  |-  ( -.  z  e.  E  <->  -.  (
z  e.  D  /\  ( g `  z
)  =/=  ( h `
 z ) ) )
19 imnan 423 . . . . . . . 8  |-  ( ( z  e.  D  ->  -.  ( g `  z
)  =/=  ( h `
 z ) )  <->  -.  ( z  e.  D  /\  ( g `  z
)  =/=  ( h `
 z ) ) )
20 nne 2605 . . . . . . . . 9  |-  ( -.  ( g `  z
)  =/=  ( h `
 z )  <->  ( g `  z )  =  ( h `  z ) )
2120imbi2i 313 . . . . . . . 8  |-  ( ( z  e.  D  ->  -.  ( g `  z
)  =/=  ( h `
 z ) )  <-> 
( z  e.  D  ->  ( g `  z
)  =  ( h `
 z ) ) )
2218, 19, 213bitr2i 276 . . . . . . 7  |-  ( -.  z  e.  E  <->  ( z  e.  D  ->  ( g `
 z )  =  ( h `  z
) ) )
2313, 22syl6ib 229 . . . . . 6  |-  ( ps 
->  ( z  e.  pred ( x ,  A ,  R )  ->  (
z  e.  D  -> 
( g `  z
)  =  ( h `
 z ) ) ) )
249, 23mpdd 41 . . . . 5  |-  ( ps 
->  ( z  e.  pred ( x ,  A ,  R )  ->  (
g `  z )  =  ( h `  z ) ) )
2524imp 430 . . . 4  |-  ( ( ps  /\  z  e. 
pred ( x ,  A ,  R ) )  ->  ( g `  z )  =  ( h `  z ) )
26 fvres 5839 . . . . . 6  |-  ( z  e.  D  ->  (
( g  |`  D ) `
 z )  =  ( g `  z
) )
279, 26syl6 34 . . . . 5  |-  ( ps 
->  ( z  e.  pred ( x ,  A ,  R )  ->  (
( g  |`  D ) `
 z )  =  ( g `  z
) ) )
2827imp 430 . . . 4  |-  ( ( ps  /\  z  e. 
pred ( x ,  A ,  R ) )  ->  ( (
g  |`  D ) `  z )  =  ( g `  z ) )
29 fvres 5839 . . . . . 6  |-  ( z  e.  D  ->  (
( h  |`  D ) `
 z )  =  ( h `  z
) )
309, 29syl6 34 . . . . 5  |-  ( ps 
->  ( z  e.  pred ( x ,  A ,  R )  ->  (
( h  |`  D ) `
 z )  =  ( h `  z
) ) )
3130imp 430 . . . 4  |-  ( ( ps  /\  z  e. 
pred ( x ,  A ,  R ) )  ->  ( (
h  |`  D ) `  z )  =  ( h `  z ) )
3225, 28, 313eqtr4d 2472 . . 3  |-  ( ( ps  /\  z  e. 
pred ( x ,  A ,  R ) )  ->  ( (
g  |`  D ) `  z )  =  ( ( h  |`  D ) `
 z ) )
3332ralrimiva 2779 . 2  |-  ( ps 
->  A. z  e.  pred  ( x ,  A ,  R ) ( ( g  |`  D ) `  z )  =  ( ( h  |`  D ) `
 z ) )
348resabs1d 5096 . . . 4  |-  ( ps 
->  ( ( g  |`  D )  |`  pred (
x ,  A ,  R ) )  =  ( g  |`  pred (
x ,  A ,  R ) ) )
358resabs1d 5096 . . . 4  |-  ( ps 
->  ( ( h  |`  D )  |`  pred (
x ,  A ,  R ) )  =  ( h  |`  pred (
x ,  A ,  R ) ) )
3634, 35eqeq12d 2443 . . 3  |-  ( ps 
->  ( ( ( g  |`  D )  |`  pred (
x ,  A ,  R ) )  =  ( ( h  |`  D )  |`  pred (
x ,  A ,  R ) )  <->  ( g  |` 
pred ( x ,  A ,  R ) )  =  ( h  |`  pred ( x ,  A ,  R ) ) ) )
371, 2, 3, 4, 5, 6, 7bnj1256 29776 . . . . . . 7  |-  ( ph  ->  E. d  e.  B  g  Fn  d )
384bnj1292 29579 . . . . . . . . 9  |-  D  C_  dom  g
39 fndm 5636 . . . . . . . . 9  |-  ( g  Fn  d  ->  dom  g  =  d )
4038, 39syl5sseq 3455 . . . . . . . 8  |-  ( g  Fn  d  ->  D  C_  d )
41 fnssres 5650 . . . . . . . 8  |-  ( ( g  Fn  d  /\  D  C_  d )  -> 
( g  |`  D )  Fn  D )
4240, 41mpdan 672 . . . . . . 7  |-  ( g  Fn  d  ->  (
g  |`  D )  Fn  D )
4337, 42bnj31 29477 . . . . . 6  |-  ( ph  ->  E. d  e.  B  ( g  |`  D )  Fn  D )
4443bnj1265 29576 . . . . 5  |-  ( ph  ->  ( g  |`  D )  Fn  D )
457, 44bnj835 29522 . . . 4  |-  ( ps 
->  ( g  |`  D )  Fn  D )
461, 2, 3, 4, 5, 6, 7bnj1259 29777 . . . . . . 7  |-  ( ph  ->  E. d  e.  B  h  Fn  d )
474bnj1293 29580 . . . . . . . . 9  |-  D  C_  dom  h
48 fndm 5636 . . . . . . . . 9  |-  ( h  Fn  d  ->  dom  h  =  d )
4947, 48syl5sseq 3455 . . . . . . . 8  |-  ( h  Fn  d  ->  D  C_  d )
50 fnssres 5650 . . . . . . . 8  |-  ( ( h  Fn  d  /\  D  C_  d )  -> 
( h  |`  D )  Fn  D )
5149, 50mpdan 672 . . . . . . 7  |-  ( h  Fn  d  ->  (
h  |`  D )  Fn  D )
5246, 51bnj31 29477 . . . . . 6  |-  ( ph  ->  E. d  e.  B  ( h  |`  D )  Fn  D )
5352bnj1265 29576 . . . . 5  |-  ( ph  ->  ( h  |`  D )  Fn  D )
547, 53bnj835 29522 . . . 4  |-  ( ps 
->  ( h  |`  D )  Fn  D )
55 fvreseq 5943 . . . 4  |-  ( ( ( ( g  |`  D )  Fn  D  /\  ( h  |`  D )  Fn  D )  /\  pred ( x ,  A ,  R )  C_  D
)  ->  ( (
( g  |`  D )  |`  pred ( x ,  A ,  R ) )  =  ( ( h  |`  D )  |` 
pred ( x ,  A ,  R ) )  <->  A. z  e.  pred  ( x ,  A ,  R ) ( ( g  |`  D ) `  z )  =  ( ( h  |`  D ) `
 z ) ) )
5645, 54, 8, 55syl21anc 1263 . . 3  |-  ( ps 
->  ( ( ( g  |`  D )  |`  pred (
x ,  A ,  R ) )  =  ( ( h  |`  D )  |`  pred (
x ,  A ,  R ) )  <->  A. z  e.  pred  ( x ,  A ,  R ) ( ( g  |`  D ) `  z
)  =  ( ( h  |`  D ) `  z ) ) )
5736, 56bitr3d 258 . 2  |-  ( ps 
->  ( ( g  |`  pred ( x ,  A ,  R ) )  =  ( h  |`  pred (
x ,  A ,  R ) )  <->  A. z  e.  pred  ( x ,  A ,  R ) ( ( g  |`  D ) `  z
)  =  ( ( h  |`  D ) `  z ) ) )
5833, 57mpbird 235 1  |-  ( ps 
->  ( g  |`  pred (
x ,  A ,  R ) )  =  ( h  |`  pred (
x ,  A ,  R ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437    e. wcel 1872   {cab 2414    =/= wne 2599   A.wral 2714   E.wrex 2715   {crab 2718    i^i cin 3378    C_ wss 3379   (/)c0 3704   <.cop 3947   class class class wbr 4366   dom cdm 4796    |` cres 4798    Fn wfn 5539   ` cfv 5544    /\ w-bnj17 29443    predc-bnj14 29445    FrSe w-bnj15 29449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-fv 5552  df-bnj17 29444
This theorem is referenced by:  bnj1311  29785
  Copyright terms: Public domain W3C validator