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Theorem bnj1311 28427
Description: Technical lemma for bnj60 28465. 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 227 . . . . . . . 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 28209 . . . . . . 7  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  R  FrSe  A )
3 ssrab2 3258 . . . . . . . 8  |-  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  C_  D
4 bnj1311.4 . . . . . . . . 9  |-  D  =  ( dom  g  i^i 
dom  h )
51bnj1235 28210 . . . . . . . . . . 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 2283 . . . . . . . . . . . 12  |-  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.  =  <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >.
9 eqid 2283 . . . . . . . . . . . 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 28416 . . . . . . . . . . 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 2373 . . . . . . . . . 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 2271 . . . . . . . . . . . . . 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 28212 . . . . . . . . . . . . 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 28200 . . . . . . . . . . . 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 2390 . . . . . . . . . . . . . 14  |-  ( d  e.  B  <->  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
)
1716simplbi 446 . . . . . . . . . . . . 13  |-  ( d  e.  B  ->  d  C_  A )
18 fndm 5343 . . . . . . . . . . . . 13  |-  ( g  Fn  d  ->  dom  g  =  d )
1917, 18bnj1241 28213 . . . . . . . . . . . 12  |-  ( ( d  e.  B  /\  g  Fn  d )  ->  dom  g  C_  A
)
2014, 19bnj593 28147 . . . . . . . . . . 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 28176 . . . . . . . . . 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 3397 . . . . . . . . . 10  |-  ( dom  g  C_  A  ->  ( dom  g  i^i  dom  h )  C_  A
)
2311, 21, 223syl 18 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  -> 
( dom  g  i^i  dom  h )  C_  A
)
244, 23syl5eqss 3222 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  D  C_  A )
253, 24syl5ss 3190 . . . . . . 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 2283 . . . . . . . 8  |-  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  =  {
x  e.  D  | 
( g `  x
)  =/=  ( h `
 x ) }
27 biid 227 . . . . . . . 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 28420 . . . . . . 7  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  =/=  (/) )
29 nfrab1 2720 . . . . . . . . 9  |-  F/_ x { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
3029nfcrii 2412 . . . . . . . 8  |-  ( z  e.  { x  e.  D  |  ( g `
 x )  =/=  ( h `  x
) }  ->  A. x  z  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) } )
3130bnj1228 28414 . . . . . . 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 1182 . . . . . 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-17 1603 . . . . . . 7  |-  ( R 
FrSe  A  ->  A. x  R  FrSe  A )
3415bnj1309 28425 . . . . . . . . . 10  |-  ( w  e.  B  ->  A. x  w  e.  B )
357, 34bnj1307 28426 . . . . . . . . 9  |-  ( w  e.  C  ->  A. x  w  e.  C )
3635nfcii 2410 . . . . . . . 8  |-  F/_ x C
3736nfcrii 2412 . . . . . . 7  |-  ( g  e.  C  ->  A. x  g  e.  C )
3836nfcrii 2412 . . . . . . 7  |-  ( h  e.  C  ->  A. x  h  e.  C )
39 ax-17 1603 . . . . . . 7  |-  ( ( g  |`  D )  =/=  ( h  |`  D )  ->  A. x ( g  |`  D )  =/=  (
h  |`  D ) )
4033, 37, 38, 39bnj982 28183 . . . . . 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 ) ) )
4132, 27, 40bnj1521 28256 . . . . 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 ) )
42 simp2 956 . . . . 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
) } )
4315, 6, 7, 4, 26, 1, 27bnj1279 28421 . . . . . . . . 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
) } )  =  (/) )
44433adant1 973 . . . . . . . 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
) } )  =  (/) )
4515, 6, 7, 4, 26, 1, 27, 44bnj1280 28423 . . . . . . 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 ) ) )
46 eqid 2283 . . . . . . 7  |-  <. x ,  ( h  |`  pred ( x ,  A ,  R ) ) >.  =  <. x ,  ( h  |`  pred ( x ,  A ,  R
) ) >.
47 eqid 2283 . . . . . . 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 ) ) >.
) ) }
4815, 6, 7, 4, 26, 1, 27, 45, 8, 9, 46, 47bnj1296 28424 . . . . . 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 ) )
4926bnj1538 28260 . . . . . . 7  |-  ( x  e.  { x  e.  D  |  ( g `
 x )  =/=  ( h `  x
) }  ->  (
g `  x )  =/=  ( h `  x
) )
5049necon2bi 2492 . . . . . 6  |-  ( ( g `  x )  =  ( h `  x )  ->  -.  x  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) } )
5148, 50syl 15 . . . . 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
) } )
5241, 42, 51bnj1304 28225 . . . 4  |-  -.  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  (
g  |`  D )  =/=  ( h  |`  D ) )
53 df-bnj17 28085 . . . 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 ) ) )
5452, 53mtbi 289 . . 3  |-  -.  (
( R  FrSe  A  /\  g  e.  C  /\  h  e.  C
)  /\  ( g  |`  D )  =/=  (
h  |`  D ) )
5554imnani 412 . 2  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  -.  ( g  |`  D )  =/=  (
h  |`  D ) )
56 nne 2450 . 2  |-  ( -.  ( g  |`  D )  =/=  ( h  |`  D )  <->  ( g  |`  D )  =  ( h  |`  D )
)
5755, 56sylib 188 1  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  ( g  |`  D )  =  ( h  |`  D ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547    i^i cin 3151    C_ wss 3152   (/)c0 3455   <.cop 3643   class class class wbr 4023   dom cdm 4689    |` cres 4691    Fn wfn 5250   ` cfv 5255    /\ w-bnj17 28084    predc-bnj14 28086    FrSe w-bnj15 28090
This theorem is referenced by:  bnj1326  28429  bnj60  28465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-bnj17 28085  df-bnj14 28087  df-bnj13 28089  df-bnj15 28091  df-bnj18 28093  df-bnj19 28095
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