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Theorem bnj1280 29389
Description: Technical lemma for bnj60 29431. 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 29388 . . . . . . 7  |-  ( ps 
->  pred ( x ,  A ,  R ) 
C_  D )
98sseld 3347 . . . . . 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 3670 . . . . . . . . 9  |-  ( ( 
pred ( x ,  A ,  R )  i^i  E )  =  (/) 
<-> 
A. z ( z  e.  pred ( x ,  A ,  R )  ->  -.  z  e.  E ) )
1210, 11sylib 189 . . . . . . . 8  |-  ( ps 
->  A. z ( z  e.  pred ( x ,  A ,  R )  ->  -.  z  e.  E ) )
131219.21bi 1774 . . . . . . 7  |-  ( ps 
->  ( z  e.  pred ( x ,  A ,  R )  ->  -.  z  e.  E )
)
14 fveq2 5728 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
g `  x )  =  ( g `  z ) )
15 fveq2 5728 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
h `  x )  =  ( h `  z ) )
1614, 15neeq12d 2616 . . . . . . . . . 10  |-  ( x  =  z  ->  (
( g `  x
)  =/=  ( h `
 x )  <->  ( g `  z )  =/=  (
h `  z )
) )
1716, 5elrab2 3094 . . . . . . . . 9  |-  ( z  e.  E  <->  ( z  e.  D  /\  (
g `  z )  =/=  ( h `  z
) ) )
1817notbii 288 . . . . . . . 8  |-  ( -.  z  e.  E  <->  -.  (
z  e.  D  /\  ( g `  z
)  =/=  ( h `
 z ) ) )
19 imnan 412 . . . . . . . 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 304 . . . . . . . 8  |-  ( ( z  e.  D  ->  -.  ( g `  z
)  =/=  ( h `
 z ) )  <-> 
( z  e.  D  ->  ( g `  z
)  =  ( h `
 z ) ) )
2218, 19, 213bitr2i 265 . . . . . . 7  |-  ( -.  z  e.  E  <->  ( z  e.  D  ->  ( g `
 z )  =  ( h `  z
) ) )
2313, 22syl6ib 218 . . . . . 6  |-  ( ps 
->  ( z  e.  pred ( x ,  A ,  R )  ->  (
z  e.  D  -> 
( g `  z
)  =  ( h `
 z ) ) ) )
249, 23mpdd 38 . . . . 5  |-  ( ps 
->  ( z  e.  pred ( x ,  A ,  R )  ->  (
g `  z )  =  ( h `  z ) ) )
2524imp 419 . . . 4  |-  ( ( ps  /\  z  e. 
pred ( x ,  A ,  R ) )  ->  ( g `  z )  =  ( h `  z ) )
26 fvres 5745 . . . . . 6  |-  ( z  e.  D  ->  (
( g  |`  D ) `
 z )  =  ( g `  z
) )
279, 26syl6 31 . . . . 5  |-  ( ps 
->  ( z  e.  pred ( x ,  A ,  R )  ->  (
( g  |`  D ) `
 z )  =  ( g `  z
) ) )
2827imp 419 . . . 4  |-  ( ( ps  /\  z  e. 
pred ( x ,  A ,  R ) )  ->  ( (
g  |`  D ) `  z )  =  ( g `  z ) )
29 fvres 5745 . . . . . 6  |-  ( z  e.  D  ->  (
( h  |`  D ) `
 z )  =  ( h `  z
) )
309, 29syl6 31 . . . . 5  |-  ( ps 
->  ( z  e.  pred ( x ,  A ,  R )  ->  (
( h  |`  D ) `
 z )  =  ( h `  z
) ) )
3130imp 419 . . . 4  |-  ( ( ps  /\  z  e. 
pred ( x ,  A ,  R ) )  ->  ( (
h  |`  D ) `  z )  =  ( h `  z ) )
3225, 28, 313eqtr4d 2478 . . 3  |-  ( ( ps  /\  z  e. 
pred ( x ,  A ,  R ) )  ->  ( (
g  |`  D ) `  z )  =  ( ( h  |`  D ) `
 z ) )
3332ralrimiva 2789 . 2  |-  ( ps 
->  A. z  e.  pred  ( x ,  A ,  R ) ( ( g  |`  D ) `  z )  =  ( ( h  |`  D ) `
 z ) )
34 resabs1 5175 . . . . 5  |-  (  pred ( x ,  A ,  R )  C_  D  ->  ( ( g  |`  D )  |`  pred (
x ,  A ,  R ) )  =  ( g  |`  pred (
x ,  A ,  R ) ) )
358, 34syl 16 . . . 4  |-  ( ps 
->  ( ( g  |`  D )  |`  pred (
x ,  A ,  R ) )  =  ( g  |`  pred (
x ,  A ,  R ) ) )
36 resabs1 5175 . . . . 5  |-  (  pred ( x ,  A ,  R )  C_  D  ->  ( ( h  |`  D )  |`  pred (
x ,  A ,  R ) )  =  ( h  |`  pred (
x ,  A ,  R ) ) )
378, 36syl 16 . . . 4  |-  ( ps 
->  ( ( h  |`  D )  |`  pred (
x ,  A ,  R ) )  =  ( h  |`  pred (
x ,  A ,  R ) ) )
3835, 37eqeq12d 2450 . . 3  |-  ( ps 
->  ( ( ( g  |`  D )  |`  pred (
x ,  A ,  R ) )  =  ( ( h  |`  D )  |`  pred (
x ,  A ,  R ) )  <->  ( g  |` 
pred ( x ,  A ,  R ) )  =  ( h  |`  pred ( x ,  A ,  R ) ) ) )
391, 2, 3, 4, 5, 6, 7bnj1256 29384 . . . . . . 7  |-  ( ph  ->  E. d  e.  B  g  Fn  d )
404bnj1292 29187 . . . . . . . . 9  |-  D  C_  dom  g
41 fndm 5544 . . . . . . . . 9  |-  ( g  Fn  d  ->  dom  g  =  d )
4240, 41syl5sseq 3396 . . . . . . . 8  |-  ( g  Fn  d  ->  D  C_  d )
43 fnssres 5558 . . . . . . . 8  |-  ( ( g  Fn  d  /\  D  C_  d )  -> 
( g  |`  D )  Fn  D )
4442, 43mpdan 650 . . . . . . 7  |-  ( g  Fn  d  ->  (
g  |`  D )  Fn  D )
4539, 44bnj31 29084 . . . . . 6  |-  ( ph  ->  E. d  e.  B  ( g  |`  D )  Fn  D )
4645bnj1265 29184 . . . . 5  |-  ( ph  ->  ( g  |`  D )  Fn  D )
477, 46bnj835 29128 . . . 4  |-  ( ps 
->  ( g  |`  D )  Fn  D )
481, 2, 3, 4, 5, 6, 7bnj1259 29385 . . . . . . 7  |-  ( ph  ->  E. d  e.  B  h  Fn  d )
494bnj1293 29188 . . . . . . . . 9  |-  D  C_  dom  h
50 fndm 5544 . . . . . . . . 9  |-  ( h  Fn  d  ->  dom  h  =  d )
5149, 50syl5sseq 3396 . . . . . . . 8  |-  ( h  Fn  d  ->  D  C_  d )
52 fnssres 5558 . . . . . . . 8  |-  ( ( h  Fn  d  /\  D  C_  d )  -> 
( h  |`  D )  Fn  D )
5351, 52mpdan 650 . . . . . . 7  |-  ( h  Fn  d  ->  (
h  |`  D )  Fn  D )
5448, 53bnj31 29084 . . . . . 6  |-  ( ph  ->  E. d  e.  B  ( h  |`  D )  Fn  D )
5554bnj1265 29184 . . . . 5  |-  ( ph  ->  ( h  |`  D )  Fn  D )
567, 55bnj835 29128 . . . 4  |-  ( ps 
->  ( h  |`  D )  Fn  D )
57 fvreseq 5833 . . . 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 ) ) )
5847, 56, 8, 57syl21anc 1183 . . 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 ) ) )
5938, 58bitr3d 247 . 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 ) ) )
6033, 59mpbird 224 1  |-  ( ps 
->  ( g  |`  pred (
x ,  A ,  R ) )  =  ( h  |`  pred (
x ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549    = wceq 1652    e. wcel 1725   {cab 2422    =/= wne 2599   A.wral 2705   E.wrex 2706   {crab 2709    i^i cin 3319    C_ wss 3320   (/)c0 3628   <.cop 3817   class class class wbr 4212   dom cdm 4878    |` cres 4880    Fn wfn 5449   ` cfv 5454    /\ w-bnj17 29050    predc-bnj14 29052    FrSe w-bnj15 29056
This theorem is referenced by:  bnj1311  29393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462  df-bnj17 29051
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