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Theorem bnj1311 29393
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
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 228 . . . . . . . 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 29175 . . . . . . 7  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  R  FrSe  A )
3 ssrab2 3428 . . . . . . . 8  |-  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  C_  D
4 bnj1311.4 . . . . . . . . 9  |-  D  =  ( dom  g  i^i 
dom  h )
51bnj1235 29176 . . . . . . . . . . 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 2436 . . . . . . . . . . . 12  |-  <. x ,  ( g  |`  pred ( x ,  A ,  R ) ) >.  =  <. x ,  ( g  |`  pred ( x ,  A ,  R
) ) >.
9 eqid 2436 . . . . . . . . . . . 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 29382 . . . . . . . . . . 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 2526 . . . . . . . . . 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 2424 . . . . . . . . . . . . . 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 29178 . . . . . . . . . . . . 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 29166 . . . . . . . . . . . 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 2543 . . . . . . . . . . . . . 14  |-  ( d  e.  B  <->  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
)
1716simplbi 447 . . . . . . . . . . . . 13  |-  ( d  e.  B  ->  d  C_  A )
18 fndm 5544 . . . . . . . . . . . . 13  |-  ( g  Fn  d  ->  dom  g  =  d )
1917, 18bnj1241 29179 . . . . . . . . . . . 12  |-  ( ( d  e.  B  /\  g  Fn  d )  ->  dom  g  C_  A
)
2014, 19bnj593 29113 . . . . . . . . . . 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 29142 . . . . . . . . . 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 3569 . . . . . . . . . 10  |-  ( dom  g  C_  A  ->  ( dom  g  i^i  dom  h )  C_  A
)
2311, 21, 223syl 19 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  -> 
( dom  g  i^i  dom  h )  C_  A
)
244, 23syl5eqss 3392 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  D  C_  A )
253, 24syl5ss 3359 . . . . . . 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 2436 . . . . . . . 8  |-  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) }  =  {
x  e.  D  | 
( g `  x
)  =/=  ( h `
 x ) }
27 biid 228 . . . . . . . 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 29386 . . . . . . 7  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) )  ->  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  =/=  (/) )
29 nfrab1 2888 . . . . . . . . 9  |-  F/_ x { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
3029nfcrii 2565 . . . . . . . 8  |-  ( z  e.  { x  e.  D  |  ( g `
 x )  =/=  ( h `  x
) }  ->  A. x  z  e.  { x  e.  D  |  (
g `  x )  =/=  ( h `  x
) } )
3130bnj1228 29380 . . . . . . 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 1184 . . . . . 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 1626 . . . . . . 7  |-  ( R 
FrSe  A  ->  A. x  R  FrSe  A )
3415bnj1309 29391 . . . . . . . . 9  |-  ( w  e.  B  ->  A. x  w  e.  B )
357, 34bnj1307 29392 . . . . . . . 8  |-  ( w  e.  C  ->  A. x  w  e.  C )
3635hblem 2540 . . . . . . 7  |-  ( g  e.  C  ->  A. x  g  e.  C )
3735hblem 2540 . . . . . . 7  |-  ( h  e.  C  ->  A. x  h  e.  C )
38 ax-17 1626 . . . . . . 7  |-  ( ( g  |`  D )  =/=  ( h  |`  D )  ->  A. x ( g  |`  D )  =/=  (
h  |`  D ) )
3933, 36, 37, 38bnj982 29149 . . . . . 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 29222 . . . . 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 958 . . . . 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 29387 . . . . . . . . 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 975 . . . . . . . 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 29389 . . . . . . 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 2436 . . . . . . 7  |-  <. x ,  ( h  |`  pred ( x ,  A ,  R ) ) >.  =  <. x ,  ( h  |`  pred ( x ,  A ,  R
) ) >.
46 eqid 2436 . . . . . . 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 29390 . . . . . 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 29226 . . . . . . 7  |-  ( x  e.  { x  e.  D  |  ( g `
 x )  =/=  ( h `  x
) }  ->  (
g `  x )  =/=  ( h `  x
) )
4948necon2bi 2650 . . . . . 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 29191 . . . 4  |-  -.  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  (
g  |`  D )  =/=  ( h  |`  D ) )
52 df-bnj17 29051 . . . 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 290 . . 3  |-  -.  (
( R  FrSe  A  /\  g  e.  C  /\  h  e.  C
)  /\  ( g  |`  D )  =/=  (
h  |`  D ) )
5453imnani 413 . 2  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  -.  ( g  |`  D )  =/=  (
h  |`  D ) )
55 nne 2605 . 2  |-  ( -.  ( g  |`  D )  =/=  ( h  |`  D )  <->  ( g  |`  D )  =  ( h  |`  D )
)
5654, 55sylib 189 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 359    /\ w3a 936    = 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:  bnj1326  29395  bnj60  29431
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-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-reg 7560  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-reu 2712  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-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  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-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-bnj17 29051  df-bnj14 29053  df-bnj13 29055  df-bnj15 29057  df-bnj18 29059  df-bnj19 29061
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