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Theorem bnj1523 33995
Description: Technical lemma for bnj1522 33996. 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 . . . . . . . . . . . . 13  |-  ( th  <->  ( ch  /\  y  e.  D  /\  A. z  e.  D  -.  z R y ) )
4 bnj1523.7 . . . . . . . . . . . . . 14  |-  ( ch  <->  ( ps  /\  x  e.  A  /\  ( F `
 x )  =/=  ( H `  x
) ) )
5 bnj1523.1 . . . . . . . . . . . . . . . . 17  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
6 bnj1523.2 . . . . . . . . . . . . . . . . 17  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
7 bnj1523.3 . . . . . . . . . . . . . . . . 17  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
8 bnj1523.4 . . . . . . . . . . . . . . . . 17  |-  F  = 
U. C
95, 6, 7, 8bnj60 33986 . . . . . . . . . . . . . . . 16  |-  ( R 
FrSe  A  ->  F  Fn  A )
101, 9bnj835 33685 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F  Fn  A )
112, 10bnj832 33683 . . . . . . . . . . . . . 14  |-  ( ps 
->  F  Fn  A
)
124, 11bnj835 33685 . . . . . . . . . . . . 13  |-  ( ch 
->  F  Fn  A
)
133, 12bnj835 33685 . . . . . . . . . . . 12  |-  ( th 
->  F  Fn  A
)
141simp2bi 1013 . . . . . . . . . . . . . . 15  |-  ( ph  ->  H  Fn  A )
152, 14bnj832 33683 . . . . . . . . . . . . . 14  |-  ( ps 
->  H  Fn  A
)
164, 15bnj835 33685 . . . . . . . . . . . . 13  |-  ( ch 
->  H  Fn  A
)
173, 16bnj835 33685 . . . . . . . . . . . 12  |-  ( th 
->  H  Fn  A
)
18 bnj213 33808 . . . . . . . . . . . . 13  |-  pred (
y ,  A ,  R )  C_  A
1918a1i 11 . . . . . . . . . . . 12  |-  ( th 
->  pred ( y ,  A ,  R ) 
C_  A )
203simp3bi 1014 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  A. z  e.  D  -.  z R y )
2120bnj1211 33724 . . . . . . . . . . . . . . . 16  |-  ( th 
->  A. z ( z  e.  D  ->  -.  z R y ) )
22 con2b 334 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  D  ->  -.  z R y )  <-> 
( z R y  ->  -.  z  e.  D ) )
2322albii 1627 . . . . . . . . . . . . . . . 16  |-  ( A. z ( z  e.  D  ->  -.  z R y )  <->  A. z
( z R y  ->  -.  z  e.  D ) )
2421, 23sylib 196 . . . . . . . . . . . . . . 15  |-  ( th 
->  A. z ( z R y  ->  -.  z  e.  D )
)
25 bnj1418 33964 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  pred ( y ,  A ,  R )  ->  z R y )
2625imim1i 58 . . . . . . . . . . . . . . . 16  |-  ( ( z R y  ->  -.  z  e.  D
)  ->  ( z  e.  pred ( y ,  A ,  R )  ->  -.  z  e.  D ) )
2726alimi 1620 . . . . . . . . . . . . . . 15  |-  ( A. z ( z R y  ->  -.  z  e.  D )  ->  A. z
( z  e.  pred ( y ,  A ,  R )  ->  -.  z  e.  D )
)
2824, 27syl 16 . . . . . . . . . . . . . 14  |-  ( th 
->  A. z ( z  e.  pred ( y ,  A ,  R )  ->  -.  z  e.  D ) )
2928bnj1142 33716 . . . . . . . . . . . . 13  |-  ( th 
->  A. z  e.  pred  ( y ,  A ,  R )  -.  z  e.  D )
30 bnj1523.8 . . . . . . . . . . . . . 14  |-  D  =  { x  e.  A  |  ( F `  x )  =/=  ( H `  x ) }
315bnj1309 33946 . . . . . . . . . . . . . . . . . . 19  |-  ( w  e.  B  ->  A. x  w  e.  B )
327, 31bnj1307 33947 . . . . . . . . . . . . . . . . . 18  |-  ( w  e.  C  ->  A. x  w  e.  C )
3332nfcii 2595 . . . . . . . . . . . . . . . . 17  |-  F/_ x C
3433nfuni 4240 . . . . . . . . . . . . . . . 16  |-  F/_ x U. C
358, 34nfcxfr 2603 . . . . . . . . . . . . . . 15  |-  F/_ x F
3635nfcrii 2597 . . . . . . . . . . . . . 14  |-  ( w  e.  F  ->  A. x  w  e.  F )
3730, 36bnj1534 33779 . . . . . . . . . . . . 13  |-  D  =  { z  e.  A  |  ( F `  z )  =/=  ( H `  z ) }
3829, 18, 37bnj1533 33778 . . . . . . . . . . . 12  |-  ( th 
->  A. z  e.  pred  ( y ,  A ,  R ) ( F `
 z )  =  ( H `  z
) )
3913, 17, 19, 38bnj1536 33780 . . . . . . . . . . 11  |-  ( th 
->  ( F  |`  pred (
y ,  A ,  R ) )  =  ( H  |`  pred (
y ,  A ,  R ) ) )
4039opeq2d 4209 . . . . . . . . . 10  |-  ( th 
->  <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >.  =  <. y ,  ( H  |`  pred ( y ,  A ,  R ) ) >.
)
4140fveq2d 5860 . . . . . . . . 9  |-  ( th 
->  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >.
)  =  ( G `
 <. y ,  ( H  |`  pred ( y ,  A ,  R
) ) >. )
)
425, 6, 7, 8bnj1500 33992 . . . . . . . . . . . . . . 15  |-  ( R 
FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
431, 42bnj835 33685 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. ) )
442, 43bnj832 33683 . . . . . . . . . . . . 13  |-  ( ps 
->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. ) )
454, 44bnj835 33685 . . . . . . . . . . . 12  |-  ( ch 
->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. ) )
4645, 36bnj1529 33994 . . . . . . . . . . 11  |-  ( ch 
->  A. y  e.  A  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) )
473, 46bnj835 33685 . . . . . . . . . 10  |-  ( th 
->  A. y  e.  A  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) )
4830bnj21 33638 . . . . . . . . . . 11  |-  D  C_  A
493simp2bi 1013 . . . . . . . . . . 11  |-  ( th 
->  y  e.  D
)
5048, 49bnj1213 33725 . . . . . . . . . 10  |-  ( th 
->  y  e.  A
)
5147, 50bnj1294 33744 . . . . . . . . 9  |-  ( th 
->  ( F `  y
)  =  ( G `
 <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >. )
)
521simp3bi 1014 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )
532, 52bnj832 33683 . . . . . . . . . . . . 13  |-  ( ps 
->  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )
544, 53bnj835 33685 . . . . . . . . . . . 12  |-  ( ch 
->  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )
55 ax-5 1691 . . . . . . . . . . . 12  |-  ( v  e.  H  ->  A. x  v  e.  H )
5654, 55bnj1529 33994 . . . . . . . . . . 11  |-  ( ch 
->  A. y  e.  A  ( H `  y )  =  ( G `  <. y ,  ( H  |`  pred ( y ,  A ,  R ) ) >. ) )
573, 56bnj835 33685 . . . . . . . . . 10  |-  ( th 
->  A. y  e.  A  ( H `  y )  =  ( G `  <. y ,  ( H  |`  pred ( y ,  A ,  R ) ) >. ) )
5857, 50bnj1294 33744 . . . . . . . . 9  |-  ( th 
->  ( H `  y
)  =  ( G `
 <. y ,  ( H  |`  pred ( y ,  A ,  R
) ) >. )
)
5941, 51, 583eqtr4d 2494 . . . . . . . 8  |-  ( th 
->  ( F `  y
)  =  ( H `
 y ) )
6030, 36bnj1534 33779 . . . . . . . . . . 11  |-  D  =  { y  e.  A  |  ( F `  y )  =/=  ( H `  y ) }
6160bnj1538 33781 . . . . . . . . . 10  |-  ( y  e.  D  ->  ( F `  y )  =/=  ( H `  y
) )
623, 61bnj836 33686 . . . . . . . . 9  |-  ( th 
->  ( F `  y
)  =/=  ( H `
 y ) )
6362neneqd 2645 . . . . . . . 8  |-  ( th 
->  -.  ( F `  y )  =  ( H `  y ) )
6459, 63pm2.65i 173 . . . . . . 7  |-  -.  th
6564nex 1614 . . . . . 6  |-  -.  E. y th
661simp1bi 1012 . . . . . . . . . 10  |-  ( ph  ->  R  FrSe  A )
672, 66bnj832 33683 . . . . . . . . 9  |-  ( ps 
->  R  FrSe  A )
684, 67bnj835 33685 . . . . . . . 8  |-  ( ch 
->  R  FrSe  A )
6948a1i 11 . . . . . . . 8  |-  ( ch 
->  D  C_  A )
704simp2bi 1013 . . . . . . . . . 10  |-  ( ch 
->  x  e.  A
)
714simp3bi 1014 . . . . . . . . . 10  |-  ( ch 
->  ( F `  x
)  =/=  ( H `
 x ) )
7230rabeq2i 3092 . . . . . . . . . 10  |-  ( x  e.  D  <->  ( x  e.  A  /\  ( F `  x )  =/=  ( H `  x
) ) )
7370, 71, 72sylanbrc 664 . . . . . . . . 9  |-  ( ch 
->  x  e.  D
)
74 ne0i 3776 . . . . . . . . 9  |-  ( x  e.  D  ->  D  =/=  (/) )
7573, 74syl 16 . . . . . . . 8  |-  ( ch 
->  D  =/=  (/) )
76 bnj69 33934 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  D  C_  A  /\  D  =/=  (/) )  ->  E. y  e.  D  A. z  e.  D  -.  z R y )
7768, 69, 75, 76syl3anc 1229 . . . . . . 7  |-  ( ch 
->  E. y  e.  D  A. z  e.  D  -.  z R y )
7877, 3bnj1209 33723 . . . . . 6  |-  ( ch 
->  E. y th )
7965, 78mto 176 . . . . 5  |-  -.  ch
8079nex 1614 . . . 4  |-  -.  E. x ch
812simprbi 464 . . . . . 6  |-  ( ps 
->  F  =/=  H
)
8211, 15, 81, 36bnj1542 33783 . . . . 5  |-  ( ps 
->  E. x  e.  A  ( F `  x )  =/=  ( H `  x ) )
835, 6, 7, 8, 1, 2bnj1525 33993 . . . . 5  |-  ( ps 
->  A. x ps )
8482, 4, 83bnj1521 33777 . . . 4  |-  ( ps 
->  E. x ch )
8580, 84mto 176 . . 3  |-  -.  ps
862, 85bnj1541 33782 . 2  |-  ( ph  ->  F  =  H )
871, 86sylbir 213 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974   A.wal 1381    = wceq 1383   E.wex 1599    e. wcel 1804   {cab 2428    =/= wne 2638   A.wral 2793   E.wrex 2794   {crab 2797    C_ wss 3461   (/)c0 3770   <.cop 4020   U.cuni 4234   class class class wbr 4437    |` cres 4991    Fn wfn 5573   ` cfv 5578    predc-bnj14 33608    FrSe w-bnj15 33612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-reg 8021  ax-inf2 8061
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-om 6686  df-1o 7132  df-bnj17 33607  df-bnj14 33609  df-bnj13 33611  df-bnj15 33613  df-bnj18 33615  df-bnj19 33617
This theorem is referenced by:  bnj1522  33996
  Copyright terms: Public domain W3C validator