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Theorem bnj1311 33168
Description: Technical lemma for bnj60 33206. 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
bnj1311.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1311.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1311.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1311.4  |-  D  =  ( dom  g  i^i 
dom  h )
Assertion
Ref Expression
bnj1311  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  ( g  |`  D )  =  ( h  |`  D ) )
Distinct variable groups:    A, d,
f, x    B, f,
g    B, h, f    D, d, x    G, d, f, g    h, G, d    R, d, f, x    g, Y    h, Y    x, g    x, h
Allowed substitution hints:    A( g, h)    B( x, d)    C( x, f, g, h, d)    D( f, g, h)    R( g, h)    G( x)    Y( x, f, d)

Proof of Theorem bnj1311
Dummy variables  w  z  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 biid 236 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  (
h  |`  D ) ) )
21bnj1232 32950 . . . . . . 7  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  R  FrSe  A )
3 ssrab2 3585 . . . . . . . 8  |-  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  C_  D
4 bnj1311.4 . . . . . . . . 9  |-  D  =  ( dom  g  i^i 
dom  h )
51bnj1235 32951 . . . . . . . . . . 11  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  -> 
g  e.  C )
6 bnj1311.2 . . . . . . . . . . . 12  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
7 bnj1311.3 . . . . . . . . . . . 12  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
8 eqid 2467 . . . . . . . . . . . 12  |-  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.  =  <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >.
9 eqid 2467 . . . . . . . . . . . 12  |-  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.
) ) }  =  { g  |  E. d  e.  B  (
g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >. ) ) }
106, 7, 8, 9bnj1234 33157 . . . . . . . . . . 11  |-  C  =  { g  |  E. d  e.  B  (
g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >. ) ) }
115, 10syl6eleq 2565 . . . . . . . . . 10  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  -> 
g  e.  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.
) ) } )
12 abid 2454 . . . . . . . . . . . . . 14  |-  ( g  e.  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )
) }  <->  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x )  =  ( G `  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.
) ) )
1312bnj1238 32953 . . . . . . . . . . . . 13  |-  ( g  e.  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )
) }  ->  E. d  e.  B  g  Fn  d )
1413bnj1196 32941 . . . . . . . . . . . 12  |-  ( g  e.  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )
) }  ->  E. d
( d  e.  B  /\  g  Fn  d
) )
15 bnj1311.1 . . . . . . . . . . . . . . 15  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
1615abeq2i 2594 . . . . . . . . . . . . . 14  |-  ( d  e.  B  <->  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
)
1716simplbi 460 . . . . . . . . . . . . 13  |-  ( d  e.  B  ->  d  C_  A )
18 fndm 5679 . . . . . . . . . . . . 13  |-  ( g  Fn  d  ->  dom  g  =  d )
1917, 18bnj1241 32954 . . . . . . . . . . . 12  |-  ( ( d  e.  B  /\  g  Fn  d )  ->  dom  g  C_  A
)
2014, 19bnj593 32890 . . . . . . . . . . 11  |-  ( g  e.  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )
) }  ->  E. d dom  g  C_  A )
2120bnj937 32918 . . . . . . . . . 10  |-  ( g  e.  { g  |  E. d  e.  B  ( g  Fn  d  /\  A. x  e.  d  ( g `  x
)  =  ( G `
 <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >. )
) }  ->  dom  g  C_  A )
22 ssinss1 3726 . . . . . . . . . 10  |-  ( dom  g  C_  A  ->  ( dom  g  i^i  dom  h )  C_  A
)
2311, 21, 223syl 20 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  -> 
( dom  g  i^i  dom  h )  C_  A
)
244, 23syl5eqss 3548 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  D  C_  A )
253, 24syl5ss 3515 . . . . . . 7  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  C_  A )
26 eqid 2467 . . . . . . . 8  |-  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  =  {
x  e.  D  | 
( g `  x
)  =/=  ( h `
 x ) }
27 biid 236 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  /\  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  /\  A. y  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  -.  y R x )  <->  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  (
g  |`  D )  =/=  ( h  |`  D ) )  /\  x  e. 
{ x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  /\  A. y  e. 
{ x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  -.  y R x ) )
2815, 6, 7, 4, 26, 1, 27bnj1253 33161 . . . . . . 7  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  =/=  (/) )
29 nfrab1 3042 . . . . . . . . 9  |-  F/_ x { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
3029nfcrii 2621 . . . . . . . 8  |-  ( z  e.  { x  e.  D  |  ( g `
 x )  =/=  ( h `  x
) }  ->  A. x  z  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) } )
3130bnj1228 33155 . . . . . . 7  |-  ( ( R  FrSe  A  /\  { x  e.  D  | 
( g `  x
)  =/=  ( h `
 x ) } 
C_  A  /\  {
x  e.  D  | 
( g `  x
)  =/=  ( h `
 x ) }  =/=  (/) )  ->  E. x  e.  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) } A. y  e.  {
x  e.  D  | 
( g `  x
)  =/=  ( h `
 x ) }  -.  y R x )
322, 25, 28, 31syl3anc 1228 . . . . . 6  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  E. x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) } A. y  e.  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  -.  y R x )
33 ax-5 1680 . . . . . . 7  |-  ( R 
FrSe  A  ->  A. x  R  FrSe  A )
3415bnj1309 33166 . . . . . . . . 9  |-  ( w  e.  B  ->  A. x  w  e.  B )
357, 34bnj1307 33167 . . . . . . . 8  |-  ( w  e.  C  ->  A. x  w  e.  C )
3635hblem 2590 . . . . . . 7  |-  ( g  e.  C  ->  A. x  g  e.  C )
3735hblem 2590 . . . . . . 7  |-  ( h  e.  C  ->  A. x  h  e.  C )
38 ax-5 1680 . . . . . . 7  |-  ( ( g  |`  D )  =/=  ( h  |`  D )  ->  A. x ( g  |`  D )  =/=  (
h  |`  D ) )
3933, 36, 37, 38bnj982 32925 . . . . . 6  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  A. x ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
4032, 27, 39bnj1521 32997 . . . . 5  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  E. x ( ( R 
FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  (
g  |`  D )  =/=  ( h  |`  D ) )  /\  x  e. 
{ x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  /\  A. y  e. 
{ x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  -.  y R x ) )
41 simp2 997 . . . . 5  |-  ( ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  /\  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  /\  A. y  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  -.  y R x )  ->  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) } )
4215, 6, 7, 4, 26, 1, 27bnj1279 33162 . . . . . . . . 9  |-  ( ( x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  /\  A. y  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  -.  y R x )  -> 
(  pred ( x ,  A ,  R )  i^i  { x  e.  D  |  ( g `
 x )  =/=  ( h `  x
) } )  =  (/) )
43423adant1 1014 . . . . . . . 8  |-  ( ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  /\  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  /\  A. y  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  -.  y R x )  -> 
(  pred ( x ,  A ,  R )  i^i  { x  e.  D  |  ( g `
 x )  =/=  ( h `  x
) } )  =  (/) )
4415, 6, 7, 4, 26, 1, 27, 43bnj1280 33164 . . . . . . 7  |-  ( ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  /\  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  /\  A. y  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  -.  y R x )  -> 
( g  |`  pred (
x ,  A ,  R ) )  =  ( h  |`  pred (
x ,  A ,  R ) ) )
45 eqid 2467 . . . . . . 7  |-  <. x ,  ( h  |`  pred ( x ,  A ,  R ) ) >.  =  <. x ,  ( h  |`  pred ( x ,  A ,  R
) ) >.
46 eqid 2467 . . . . . . 7  |-  { h  |  E. d  e.  B  ( h  Fn  d  /\  A. x  e.  d  ( h `  x
)  =  ( G `
 <. x ,  ( h  |`  pred ( x ,  A ,  R
) ) >. )
) }  =  {
h  |  E. d  e.  B  ( h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  <. x ,  ( h  |`  pred ( x ,  A ,  R ) ) >.
) ) }
4715, 6, 7, 4, 26, 1, 27, 44, 8, 9, 45, 46bnj1296 33165 . . . . . 6  |-  ( ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  /\  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  /\  A. y  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  -.  y R x )  -> 
( g `  x
)  =  ( h `
 x ) )
4826bnj1538 33001 . . . . . . 7  |-  ( x  e.  { x  e.  D  |  ( g `
 x )  =/=  ( h `  x
) }  ->  (
g `  x )  =/=  ( h `  x
) )
4948necon2bi 2704 . . . . . 6  |-  ( ( g `  x )  =  ( h `  x )  ->  -.  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) } )
5047, 49syl 16 . . . . 5  |-  ( ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  /\  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  /\  A. y  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  -.  y R x )  ->  -.  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) } )
5140, 41, 50bnj1304 32966 . . . 4  |-  -.  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  (
g  |`  D )  =/=  ( h  |`  D ) )
52 df-bnj17 32828 . . . 4  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  <->  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C )  /\  (
g  |`  D )  =/=  ( h  |`  D ) ) )
5351, 52mtbi 298 . . 3  |-  -.  (
( R  FrSe  A  /\  g  e.  C  /\  h  e.  C
)  /\  ( g  |`  D )  =/=  (
h  |`  D ) )
5453imnani 423 . 2  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  -.  ( g  |`  D )  =/=  (
h  |`  D ) )
55 nne 2668 . 2  |-  ( -.  ( g  |`  D )  =/=  ( h  |`  D )  <->  ( g  |`  D )  =  ( h  |`  D )
)
5654, 55sylib 196 1  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  ( g  |`  D )  =  ( h  |`  D ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818    i^i cin 3475    C_ wss 3476   (/)c0 3785   <.cop 4033   class class class wbr 4447   dom cdm 4999    |` cres 5001    Fn wfn 5582   ` cfv 5587    /\ w-bnj17 32827    predc-bnj14 32829    FrSe w-bnj15 32833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-reg 8017  ax-inf2 8057
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-om 6680  df-1o 7130  df-bnj17 32828  df-bnj14 32830  df-bnj13 32832  df-bnj15 32834  df-bnj18 32836  df-bnj19 32838
This theorem is referenced by:  bnj1326  33170  bnj60  33206
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