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Theorem bnj1523 29101
Description: Technical lemma for bnj1522 29102. 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
bnj1523.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1523.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1523.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1523.4  |-  F  = 
U. C
bnj1523.5  |-  ( ph  <->  ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) ) )
bnj1523.6  |-  ( ps  <->  (
ph  /\  F  =/=  H ) )
bnj1523.7  |-  ( ch  <->  ( ps  /\  x  e.  A  /\  ( F `
 x )  =/=  ( H `  x
) ) )
bnj1523.8  |-  D  =  { x  e.  A  |  ( F `  x )  =/=  ( H `  x ) }
bnj1523.9  |-  ( th  <->  ( ch  /\  y  e.  D  /\  A. z  e.  D  -.  z R y ) )
Assertion
Ref Expression
bnj1523  |-  ( ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )  ->  F  =  H )
Distinct variable groups:    A, d,
f, x    y, A, z, x    B, f    y, D, z    y, F, z    G, d, f, x    y, G    x, H, y, z    R, d, f, x    y, R, z    Y, d    ch, y
Allowed substitution hints:    ph( x, y, z, f, d)    ps( x, y, z, f, d)    ch( x, z, f, d)    th( x, y, z, f, d)    B( x, y, z, d)    C( x, y, z, f, d)    D( x, f, d)    F( x, f, d)    G( z)    H( f, d)    Y( x, y, z, f)

Proof of Theorem bnj1523
Dummy variables  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj1523.5 . 2  |-  ( ph  <->  ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) ) )
2 bnj1523.6 . . 3  |-  ( ps  <->  (
ph  /\  F  =/=  H ) )
3 bnj1523.9 . . . . . . . . . . 11  |-  ( th  <->  ( ch  /\  y  e.  D  /\  A. z  e.  D  -.  z R y ) )
4 bnj1523.7 . . . . . . . . . . . . 13  |-  ( ch  <->  ( ps  /\  x  e.  A  /\  ( F `
 x )  =/=  ( H `  x
) ) )
5 bnj1523.1 . . . . . . . . . . . . . . . 16  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
6 bnj1523.2 . . . . . . . . . . . . . . . 16  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
7 bnj1523.3 . . . . . . . . . . . . . . . 16  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
8 bnj1523.4 . . . . . . . . . . . . . . . 16  |-  F  = 
U. C
95, 6, 7, 8bnj1500 29098 . . . . . . . . . . . . . . 15  |-  ( R 
FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
101, 9bnj835 28789 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. ) )
112, 10bnj832 28787 . . . . . . . . . . . . 13  |-  ( ps 
->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. ) )
124, 11bnj835 28789 . . . . . . . . . . . 12  |-  ( ch 
->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. ) )
135bnj1309 29052 . . . . . . . . . . . . . . . . 17  |-  ( w  e.  B  ->  A. x  w  e.  B )
147, 13bnj1307 29053 . . . . . . . . . . . . . . . 16  |-  ( w  e.  C  ->  A. x  w  e.  C )
1514nfcii 2410 . . . . . . . . . . . . . . 15  |-  F/_ x C
1615nfuni 3833 . . . . . . . . . . . . . 14  |-  F/_ x U. C
178, 16nfcxfr 2416 . . . . . . . . . . . . 13  |-  F/_ x F
1817nfcrii 2412 . . . . . . . . . . . 12  |-  ( w  e.  F  ->  A. x  w  e.  F )
1912, 18bnj1529 29100 . . . . . . . . . . 11  |-  ( ch 
->  A. y  e.  A  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) )
203, 19bnj835 28789 . . . . . . . . . 10  |-  ( th 
->  A. y  e.  A  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) )
21 bnj1523.8 . . . . . . . . . . . 12  |-  D  =  { x  e.  A  |  ( F `  x )  =/=  ( H `  x ) }
2221bnj21 28743 . . . . . . . . . . 11  |-  D  C_  A
233simp2bi 971 . . . . . . . . . . 11  |-  ( th 
->  y  e.  D
)
2422, 23bnj1213 28831 . . . . . . . . . 10  |-  ( th 
->  y  e.  A
)
2520, 24bnj1294 28850 . . . . . . . . 9  |-  ( th 
->  ( F `  y
)  =  ( G `
 <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >. )
)
261simp3bi 972 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )
272, 26bnj832 28787 . . . . . . . . . . . . . 14  |-  ( ps 
->  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )
284, 27bnj835 28789 . . . . . . . . . . . . 13  |-  ( ch 
->  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )
29 ax-17 1603 . . . . . . . . . . . . 13  |-  ( v  e.  H  ->  A. x  v  e.  H )
3028, 29bnj1529 29100 . . . . . . . . . . . 12  |-  ( ch 
->  A. y  e.  A  ( H `  y )  =  ( G `  <. y ,  ( H  |`  pred ( y ,  A ,  R ) ) >. ) )
313, 30bnj835 28789 . . . . . . . . . . 11  |-  ( th 
->  A. y  e.  A  ( H `  y )  =  ( G `  <. y ,  ( H  |`  pred ( y ,  A ,  R ) ) >. ) )
3231, 24bnj1294 28850 . . . . . . . . . 10  |-  ( th 
->  ( H `  y
)  =  ( G `
 <. y ,  ( H  |`  pred ( y ,  A ,  R
) ) >. )
)
335, 6, 7, 8bnj60 29092 . . . . . . . . . . . . . . . . 17  |-  ( R 
FrSe  A  ->  F  Fn  A )
341, 33bnj835 28789 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  F  Fn  A )
352, 34bnj832 28787 . . . . . . . . . . . . . . 15  |-  ( ps 
->  F  Fn  A
)
364, 35bnj835 28789 . . . . . . . . . . . . . 14  |-  ( ch 
->  F  Fn  A
)
373, 36bnj835 28789 . . . . . . . . . . . . 13  |-  ( th 
->  F  Fn  A
)
381simp2bi 971 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  H  Fn  A )
392, 38bnj832 28787 . . . . . . . . . . . . . . 15  |-  ( ps 
->  H  Fn  A
)
404, 39bnj835 28789 . . . . . . . . . . . . . 14  |-  ( ch 
->  H  Fn  A
)
413, 40bnj835 28789 . . . . . . . . . . . . 13  |-  ( th 
->  H  Fn  A
)
42 bnj213 28914 . . . . . . . . . . . . . 14  |-  pred (
y ,  A ,  R )  C_  A
4342a1i 10 . . . . . . . . . . . . 13  |-  ( th 
->  pred ( y ,  A ,  R ) 
C_  A )
443simp3bi 972 . . . . . . . . . . . . . . . . . 18  |-  ( th 
->  A. z  e.  D  -.  z R y )
4544bnj1211 28830 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  A. z ( z  e.  D  ->  -.  z R y ) )
46 con2b 324 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  D  ->  -.  z R y )  <-> 
( z R y  ->  -.  z  e.  D ) )
4746albii 1553 . . . . . . . . . . . . . . . . 17  |-  ( A. z ( z  e.  D  ->  -.  z R y )  <->  A. z
( z R y  ->  -.  z  e.  D ) )
4845, 47sylib 188 . . . . . . . . . . . . . . . 16  |-  ( th 
->  A. z ( z R y  ->  -.  z  e.  D )
)
49 bnj1418 29070 . . . . . . . . . . . . . . . . . 18  |-  ( z  e.  pred ( y ,  A ,  R )  ->  z R y )
5049imim1i 54 . . . . . . . . . . . . . . . . 17  |-  ( ( z R y  ->  -.  z  e.  D
)  ->  ( z  e.  pred ( y ,  A ,  R )  ->  -.  z  e.  D ) )
5150alimi 1546 . . . . . . . . . . . . . . . 16  |-  ( A. z ( z R y  ->  -.  z  e.  D )  ->  A. z
( z  e.  pred ( y ,  A ,  R )  ->  -.  z  e.  D )
)
5248, 51syl 15 . . . . . . . . . . . . . . 15  |-  ( th 
->  A. z ( z  e.  pred ( y ,  A ,  R )  ->  -.  z  e.  D ) )
5352bnj1142 28821 . . . . . . . . . . . . . 14  |-  ( th 
->  A. z  e.  pred  ( y ,  A ,  R )  -.  z  e.  D )
5421, 18bnj1534 28885 . . . . . . . . . . . . . 14  |-  D  =  { z  e.  A  |  ( F `  z )  =/=  ( H `  z ) }
5553, 42, 54bnj1533 28884 . . . . . . . . . . . . 13  |-  ( th 
->  A. z  e.  pred  ( y ,  A ,  R ) ( F `
 z )  =  ( H `  z
) )
5637, 41, 43, 55bnj1536 28886 . . . . . . . . . . . 12  |-  ( th 
->  ( F  |`  pred (
y ,  A ,  R ) )  =  ( H  |`  pred (
y ,  A ,  R ) ) )
5756opeq2d 3803 . . . . . . . . . . 11  |-  ( th 
->  <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >.  =  <. y ,  ( H  |`  pred ( y ,  A ,  R ) ) >.
)
5857fveq2d 5529 . . . . . . . . . 10  |-  ( th 
->  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >.
)  =  ( G `
 <. y ,  ( H  |`  pred ( y ,  A ,  R
) ) >. )
)
5932, 58eqtr4d 2318 . . . . . . . . 9  |-  ( th 
->  ( H `  y
)  =  ( G `
 <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >. )
)
6025, 59eqtr4d 2318 . . . . . . . 8  |-  ( th 
->  ( F `  y
)  =  ( H `
 y ) )
6121, 18bnj1534 28885 . . . . . . . . . . 11  |-  D  =  { y  e.  A  |  ( F `  y )  =/=  ( H `  y ) }
6261bnj1538 28887 . . . . . . . . . 10  |-  ( y  e.  D  ->  ( F `  y )  =/=  ( H `  y
) )
633, 62bnj836 28790 . . . . . . . . 9  |-  ( th 
->  ( F `  y
)  =/=  ( H `
 y ) )
6463neneqd 2462 . . . . . . . 8  |-  ( th 
->  -.  ( F `  y )  =  ( H `  y ) )
6560, 64pm2.65i 165 . . . . . . 7  |-  -.  th
6665nex 1542 . . . . . 6  |-  -.  E. y th
671simp1bi 970 . . . . . . . . . 10  |-  ( ph  ->  R  FrSe  A )
682, 67bnj832 28787 . . . . . . . . 9  |-  ( ps 
->  R  FrSe  A )
694, 68bnj835 28789 . . . . . . . 8  |-  ( ch 
->  R  FrSe  A )
7022a1i 10 . . . . . . . 8  |-  ( ch 
->  D  C_  A )
714simp2bi 971 . . . . . . . . . 10  |-  ( ch 
->  x  e.  A
)
724simp3bi 972 . . . . . . . . . 10  |-  ( ch 
->  ( F `  x
)  =/=  ( H `
 x ) )
7321rabeq2i 2785 . . . . . . . . . 10  |-  ( x  e.  D  <->  ( x  e.  A  /\  ( F `  x )  =/=  ( H `  x
) ) )
7471, 72, 73sylanbrc 645 . . . . . . . . 9  |-  ( ch 
->  x  e.  D
)
75 ne0i 3461 . . . . . . . . 9  |-  ( x  e.  D  ->  D  =/=  (/) )
7674, 75syl 15 . . . . . . . 8  |-  ( ch 
->  D  =/=  (/) )
77 bnj69 29040 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  D  C_  A  /\  D  =/=  (/) )  ->  E. y  e.  D  A. z  e.  D  -.  z R y )
7869, 70, 76, 77syl3anc 1182 . . . . . . 7  |-  ( ch 
->  E. y  e.  D  A. z  e.  D  -.  z R y )
7978, 3bnj1209 28829 . . . . . 6  |-  ( ch 
->  E. y th )
8066, 79mto 167 . . . . 5  |-  -.  ch
8180nex 1542 . . . 4  |-  -.  E. x ch
822simprbi 450 . . . . . 6  |-  ( ps 
->  F  =/=  H
)
8335, 39, 82, 18bnj1542 28889 . . . . 5  |-  ( ps 
->  E. x  e.  A  ( F `  x )  =/=  ( H `  x ) )
845, 6, 7, 8, 1, 2bnj1525 29099 . . . . 5  |-  ( ps 
->  A. x ps )
8583, 4, 84bnj1521 28883 . . . 4  |-  ( ps 
->  E. x ch )
8681, 85mto 167 . . 3  |-  -.  ps
872, 86bnj1541 28888 . 2  |-  ( ph  ->  F  =  H )
881, 87sylbir 204 1  |-  ( ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )  ->  F  =  H )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547    C_ wss 3152   (/)c0 3455   <.cop 3643   U.cuni 3827   class class class wbr 4023    |` cres 4691    Fn wfn 5250   ` cfv 5255    predc-bnj14 28713    FrSe w-bnj15 28717
This theorem is referenced by:  bnj1522  29102
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 28712  df-bnj14 28714  df-bnj13 28716  df-bnj15 28718  df-bnj18 28720  df-bnj19 28722
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