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Theorem bnj1311 32027
Description: Technical lemma for bnj60 32065. 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 31809 . . . . . . 7  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  R  FrSe  A )
3 ssrab2 3449 . . . . . . . 8  |-  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  C_  D
4 bnj1311.4 . . . . . . . . 9  |-  D  =  ( dom  g  i^i 
dom  h )
51bnj1235 31810 . . . . . . . . . . 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 2443 . . . . . . . . . . . 12  |-  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.  =  <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >.
9 eqid 2443 . . . . . . . . . . . 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 32016 . . . . . . . . . . 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 2533 . . . . . . . . . 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 2431 . . . . . . . . . . . . . 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 31812 . . . . . . . . . . . . 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 31800 . . . . . . . . . . . 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 2554 . . . . . . . . . . . . . 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 5522 . . . . . . . . . . . . 13  |-  ( g  Fn  d  ->  dom  g  =  d )
1917, 18bnj1241 31813 . . . . . . . . . . . 12  |-  ( ( d  e.  B  /\  g  Fn  d )  ->  dom  g  C_  A
)
2014, 19bnj593 31749 . . . . . . . . . . 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 31777 . . . . . . . . . 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 3590 . . . . . . . . . 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 3412 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  D  C_  A )
253, 24syl5ss 3379 . . . . . . 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 2443 . . . . . . . 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 32020 . . . . . . 7  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  =/=  (/) )
29 nfrab1 2913 . . . . . . . . 9  |-  F/_ x { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
3029nfcrii 2581 . . . . . . . 8  |-  ( z  e.  { x  e.  D  |  ( g `
 x )  =/=  ( h `  x
) }  ->  A. x  z  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) } )
3130bnj1228 32014 . . . . . . 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 1218 . . . . . 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 1670 . . . . . . 7  |-  ( R 
FrSe  A  ->  A. x  R  FrSe  A )
3415bnj1309 32025 . . . . . . . . 9  |-  ( w  e.  B  ->  A. x  w  e.  B )
357, 34bnj1307 32026 . . . . . . . 8  |-  ( w  e.  C  ->  A. x  w  e.  C )
3635hblem 2550 . . . . . . 7  |-  ( g  e.  C  ->  A. x  g  e.  C )
3735hblem 2550 . . . . . . 7  |-  ( h  e.  C  ->  A. x  h  e.  C )
38 ax-5 1670 . . . . . . 7  |-  ( ( g  |`  D )  =/=  ( h  |`  D )  ->  A. x ( g  |`  D )  =/=  (
h  |`  D ) )
3933, 36, 37, 38bnj982 31784 . . . . . 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 31856 . . . . 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 989 . . . . 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 32021 . . . . . . . . 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 1006 . . . . . . . 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 32023 . . . . . . 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 2443 . . . . . . 7  |-  <. x ,  ( h  |`  pred ( x ,  A ,  R ) ) >.  =  <. x ,  ( h  |`  pred ( x ,  A ,  R
) ) >.
46 eqid 2443 . . . . . . 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 32024 . . . . . 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 31860 . . . . . . 7  |-  ( x  e.  { x  e.  D  |  ( g `
 x )  =/=  ( h `  x
) }  ->  (
g `  x )  =/=  ( h `  x
) )
4948necon2bi 2669 . . . . . 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 31825 . . . 4  |-  -.  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  (
g  |`  D )  =/=  ( h  |`  D ) )
52 df-bnj17 31687 . . . 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 2624 . 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 965    = wceq 1369    e. wcel 1756   {cab 2429    =/= wne 2618   A.wral 2727   E.wrex 2728   {crab 2731    i^i cin 3339    C_ wss 3340   (/)c0 3649   <.cop 3895   class class class wbr 4304   dom cdm 4852    |` cres 4854    Fn wfn 5425   ` cfv 5430    /\ w-bnj17 31686    predc-bnj14 31688    FrSe w-bnj15 31692
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-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-reg 7819  ax-inf2 7859
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  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 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-om 6489  df-1o 6932  df-bnj17 31687  df-bnj14 31689  df-bnj13 31691  df-bnj15 31693  df-bnj18 31695  df-bnj19 31697
This theorem is referenced by:  bnj1326  32029  bnj60  32065
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