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Theorem bnj1280 32011
Description: Technical lemma for bnj60 32053. 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 32010 . . . . . . 7  |-  ( ps 
->  pred ( x ,  A ,  R ) 
C_  D )
98sseld 3355 . . . . . 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 3721 . . . . . . . . 9  |-  ( ( 
pred ( x ,  A ,  R )  i^i  E )  =  (/) 
<-> 
A. z ( z  e.  pred ( x ,  A ,  R )  ->  -.  z  e.  E ) )
1210, 11sylib 196 . . . . . . . 8  |-  ( ps 
->  A. z ( z  e.  pred ( x ,  A ,  R )  ->  -.  z  e.  E ) )
131219.21bi 1804 . . . . . . 7  |-  ( ps 
->  ( z  e.  pred ( x ,  A ,  R )  ->  -.  z  e.  E )
)
14 fveq2 5691 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
g `  x )  =  ( g `  z ) )
15 fveq2 5691 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
h `  x )  =  ( h `  z ) )
1614, 15neeq12d 2623 . . . . . . . . . 10  |-  ( x  =  z  ->  (
( g `  x
)  =/=  ( h `
 x )  <->  ( g `  z )  =/=  (
h `  z )
) )
1716, 5elrab2 3119 . . . . . . . . 9  |-  ( z  e.  E  <->  ( z  e.  D  /\  (
g `  z )  =/=  ( h `  z
) ) )
1817notbii 296 . . . . . . . 8  |-  ( -.  z  e.  E  <->  -.  (
z  e.  D  /\  ( g `  z
)  =/=  ( h `
 z ) ) )
19 imnan 422 . . . . . . . 8  |-  ( ( z  e.  D  ->  -.  ( g `  z
)  =/=  ( h `
 z ) )  <->  -.  ( z  e.  D  /\  ( g `  z
)  =/=  ( h `
 z ) ) )
20 nne 2612 . . . . . . . . 9  |-  ( -.  ( g `  z
)  =/=  ( h `
 z )  <->  ( g `  z )  =  ( h `  z ) )
2120imbi2i 312 . . . . . . . 8  |-  ( ( z  e.  D  ->  -.  ( g `  z
)  =/=  ( h `
 z ) )  <-> 
( z  e.  D  ->  ( g `  z
)  =  ( h `
 z ) ) )
2218, 19, 213bitr2i 273 . . . . . . 7  |-  ( -.  z  e.  E  <->  ( z  e.  D  ->  ( g `
 z )  =  ( h `  z
) ) )
2313, 22syl6ib 226 . . . . . 6  |-  ( ps 
->  ( z  e.  pred ( x ,  A ,  R )  ->  (
z  e.  D  -> 
( g `  z
)  =  ( h `
 z ) ) ) )
249, 23mpdd 40 . . . . 5  |-  ( ps 
->  ( z  e.  pred ( x ,  A ,  R )  ->  (
g `  z )  =  ( h `  z ) ) )
2524imp 429 . . . 4  |-  ( ( ps  /\  z  e. 
pred ( x ,  A ,  R ) )  ->  ( g `  z )  =  ( h `  z ) )
26 fvres 5704 . . . . . 6  |-  ( z  e.  D  ->  (
( g  |`  D ) `
 z )  =  ( g `  z
) )
279, 26syl6 33 . . . . 5  |-  ( ps 
->  ( z  e.  pred ( x ,  A ,  R )  ->  (
( g  |`  D ) `
 z )  =  ( g `  z
) ) )
2827imp 429 . . . 4  |-  ( ( ps  /\  z  e. 
pred ( x ,  A ,  R ) )  ->  ( (
g  |`  D ) `  z )  =  ( g `  z ) )
29 fvres 5704 . . . . . 6  |-  ( z  e.  D  ->  (
( h  |`  D ) `
 z )  =  ( h `  z
) )
309, 29syl6 33 . . . . 5  |-  ( ps 
->  ( z  e.  pred ( x ,  A ,  R )  ->  (
( h  |`  D ) `
 z )  =  ( h `  z
) ) )
3130imp 429 . . . 4  |-  ( ( ps  /\  z  e. 
pred ( x ,  A ,  R ) )  ->  ( (
h  |`  D ) `  z )  =  ( h `  z ) )
3225, 28, 313eqtr4d 2485 . . 3  |-  ( ( ps  /\  z  e. 
pred ( x ,  A ,  R ) )  ->  ( (
g  |`  D ) `  z )  =  ( ( h  |`  D ) `
 z ) )
3332ralrimiva 2799 . 2  |-  ( ps 
->  A. z  e.  pred  ( x ,  A ,  R ) ( ( g  |`  D ) `  z )  =  ( ( h  |`  D ) `
 z ) )
34 resabs1 5139 . . . . 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 5139 . . . . 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 2457 . . 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 32006 . . . . . . 7  |-  ( ph  ->  E. d  e.  B  g  Fn  d )
404bnj1292 31809 . . . . . . . . 9  |-  D  C_  dom  g
41 fndm 5510 . . . . . . . . 9  |-  ( g  Fn  d  ->  dom  g  =  d )
4240, 41syl5sseq 3404 . . . . . . . 8  |-  ( g  Fn  d  ->  D  C_  d )
43 fnssres 5524 . . . . . . . 8  |-  ( ( g  Fn  d  /\  D  C_  d )  -> 
( g  |`  D )  Fn  D )
4442, 43mpdan 668 . . . . . . 7  |-  ( g  Fn  d  ->  (
g  |`  D )  Fn  D )
4539, 44bnj31 31708 . . . . . 6  |-  ( ph  ->  E. d  e.  B  ( g  |`  D )  Fn  D )
4645bnj1265 31806 . . . . 5  |-  ( ph  ->  ( g  |`  D )  Fn  D )
477, 46bnj835 31752 . . . 4  |-  ( ps 
->  ( g  |`  D )  Fn  D )
481, 2, 3, 4, 5, 6, 7bnj1259 32007 . . . . . . 7  |-  ( ph  ->  E. d  e.  B  h  Fn  d )
494bnj1293 31810 . . . . . . . . 9  |-  D  C_  dom  h
50 fndm 5510 . . . . . . . . 9  |-  ( h  Fn  d  ->  dom  h  =  d )
5149, 50syl5sseq 3404 . . . . . . . 8  |-  ( h  Fn  d  ->  D  C_  d )
52 fnssres 5524 . . . . . . . 8  |-  ( ( h  Fn  d  /\  D  C_  d )  -> 
( h  |`  D )  Fn  D )
5351, 52mpdan 668 . . . . . . 7  |-  ( h  Fn  d  ->  (
h  |`  D )  Fn  D )
5448, 53bnj31 31708 . . . . . 6  |-  ( ph  ->  E. d  e.  B  ( h  |`  D )  Fn  D )
5554bnj1265 31806 . . . . 5  |-  ( ph  ->  ( h  |`  D )  Fn  D )
567, 55bnj835 31752 . . . 4  |-  ( ps 
->  ( h  |`  D )  Fn  D )
57 fvreseq 5805 . . . 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 1217 . . 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 255 . 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 232 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 184    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369    e. wcel 1756   {cab 2429    =/= wne 2606   A.wral 2715   E.wrex 2716   {crab 2719    i^i cin 3327    C_ wss 3328   (/)c0 3637   <.cop 3883   class class class wbr 4292   dom cdm 4840    |` cres 4842    Fn wfn 5413   ` cfv 5418    /\ w-bnj17 31674    predc-bnj14 31676    FrSe w-bnj15 31680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-fv 5426  df-bnj17 31675
This theorem is referenced by:  bnj1311  32015
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