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Theorem bnj1523 29377
Description: Technical lemma for bnj1522 29378. 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 29368 . . . . . . . . . . . . . . . 16  |-  ( R 
FrSe  A  ->  F  Fn  A )
101, 9bnj835 29065 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F  Fn  A )
112, 10bnj832 29063 . . . . . . . . . . . . . 14  |-  ( ps 
->  F  Fn  A
)
124, 11bnj835 29065 . . . . . . . . . . . . 13  |-  ( ch 
->  F  Fn  A
)
133, 12bnj835 29065 . . . . . . . . . . . 12  |-  ( th 
->  F  Fn  A
)
141simp2bi 973 . . . . . . . . . . . . . . 15  |-  ( ph  ->  H  Fn  A )
152, 14bnj832 29063 . . . . . . . . . . . . . 14  |-  ( ps 
->  H  Fn  A
)
164, 15bnj835 29065 . . . . . . . . . . . . 13  |-  ( ch 
->  H  Fn  A
)
173, 16bnj835 29065 . . . . . . . . . . . 12  |-  ( th 
->  H  Fn  A
)
18 bnj213 29190 . . . . . . . . . . . . 13  |-  pred (
y ,  A ,  R )  C_  A
1918a1i 11 . . . . . . . . . . . 12  |-  ( th 
->  pred ( y ,  A ,  R ) 
C_  A )
203simp3bi 974 . . . . . . . . . . . . . . . . 17  |-  ( th 
->  A. z  e.  D  -.  z R y )
2120bnj1211 29106 . . . . . . . . . . . . . . . 16  |-  ( th 
->  A. z ( z  e.  D  ->  -.  z R y ) )
22 con2b 325 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  D  ->  -.  z R y )  <-> 
( z R y  ->  -.  z  e.  D ) )
2322albii 1575 . . . . . . . . . . . . . . . 16  |-  ( A. z ( z  e.  D  ->  -.  z R y )  <->  A. z
( z R y  ->  -.  z  e.  D ) )
2421, 23sylib 189 . . . . . . . . . . . . . . 15  |-  ( th 
->  A. z ( z R y  ->  -.  z  e.  D )
)
25 bnj1418 29346 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  pred ( y ,  A ,  R )  ->  z R y )
2625imim1i 56 . . . . . . . . . . . . . . . 16  |-  ( ( z R y  ->  -.  z  e.  D
)  ->  ( z  e.  pred ( y ,  A ,  R )  ->  -.  z  e.  D ) )
2726alimi 1568 . . . . . . . . . . . . . . 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 29097 . . . . . . . . . . . . 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 29328 . . . . . . . . . . . . . . . . . . 19  |-  ( w  e.  B  ->  A. x  w  e.  B )
327, 31bnj1307 29329 . . . . . . . . . . . . . . . . . 18  |-  ( w  e.  C  ->  A. x  w  e.  C )
3332nfcii 2562 . . . . . . . . . . . . . . . . 17  |-  F/_ x C
3433nfuni 4013 . . . . . . . . . . . . . . . 16  |-  F/_ x U. C
358, 34nfcxfr 2568 . . . . . . . . . . . . . . 15  |-  F/_ x F
3635nfcrii 2564 . . . . . . . . . . . . . 14  |-  ( w  e.  F  ->  A. x  w  e.  F )
3730, 36bnj1534 29161 . . . . . . . . . . . . 13  |-  D  =  { z  e.  A  |  ( F `  z )  =/=  ( H `  z ) }
3829, 18, 37bnj1533 29160 . . . . . . . . . . . 12  |-  ( th 
->  A. z  e.  pred  ( y ,  A ,  R ) ( F `
 z )  =  ( H `  z
) )
3913, 17, 19, 38bnj1536 29162 . . . . . . . . . . 11  |-  ( th 
->  ( F  |`  pred (
y ,  A ,  R ) )  =  ( H  |`  pred (
y ,  A ,  R ) ) )
4039opeq2d 3983 . . . . . . . . . 10  |-  ( th 
->  <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >.  =  <. y ,  ( H  |`  pred ( y ,  A ,  R ) ) >.
)
4140fveq2d 5724 . . . . . . . . 9  |-  ( th 
->  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >.
)  =  ( G `
 <. y ,  ( H  |`  pred ( y ,  A ,  R
) ) >. )
)
425, 6, 7, 8bnj1500 29374 . . . . . . . . . . . . . . 15  |-  ( R 
FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
431, 42bnj835 29065 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. ) )
442, 43bnj832 29063 . . . . . . . . . . . . 13  |-  ( ps 
->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. ) )
454, 44bnj835 29065 . . . . . . . . . . . 12  |-  ( ch 
->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. ) )
4645, 36bnj1529 29376 . . . . . . . . . . 11  |-  ( ch 
->  A. y  e.  A  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) )
473, 46bnj835 29065 . . . . . . . . . 10  |-  ( th 
->  A. y  e.  A  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) )
4830bnj21 29019 . . . . . . . . . . 11  |-  D  C_  A
493simp2bi 973 . . . . . . . . . . 11  |-  ( th 
->  y  e.  D
)
5048, 49bnj1213 29107 . . . . . . . . . 10  |-  ( th 
->  y  e.  A
)
5147, 50bnj1294 29126 . . . . . . . . 9  |-  ( th 
->  ( F `  y
)  =  ( G `
 <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >. )
)
521simp3bi 974 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )
532, 52bnj832 29063 . . . . . . . . . . . . 13  |-  ( ps 
->  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )
544, 53bnj835 29065 . . . . . . . . . . . 12  |-  ( ch 
->  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )
55 ax-17 1626 . . . . . . . . . . . 12  |-  ( v  e.  H  ->  A. x  v  e.  H )
5654, 55bnj1529 29376 . . . . . . . . . . 11  |-  ( ch 
->  A. y  e.  A  ( H `  y )  =  ( G `  <. y ,  ( H  |`  pred ( y ,  A ,  R ) ) >. ) )
573, 56bnj835 29065 . . . . . . . . . 10  |-  ( th 
->  A. y  e.  A  ( H `  y )  =  ( G `  <. y ,  ( H  |`  pred ( y ,  A ,  R ) ) >. ) )
5857, 50bnj1294 29126 . . . . . . . . 9  |-  ( th 
->  ( H `  y
)  =  ( G `
 <. y ,  ( H  |`  pred ( y ,  A ,  R
) ) >. )
)
5941, 51, 583eqtr4d 2477 . . . . . . . 8  |-  ( th 
->  ( F `  y
)  =  ( H `
 y ) )
6030, 36bnj1534 29161 . . . . . . . . . . 11  |-  D  =  { y  e.  A  |  ( F `  y )  =/=  ( H `  y ) }
6160bnj1538 29163 . . . . . . . . . 10  |-  ( y  e.  D  ->  ( F `  y )  =/=  ( H `  y
) )
623, 61bnj836 29066 . . . . . . . . 9  |-  ( th 
->  ( F `  y
)  =/=  ( H `
 y ) )
6362neneqd 2614 . . . . . . . 8  |-  ( th 
->  -.  ( F `  y )  =  ( H `  y ) )
6459, 63pm2.65i 167 . . . . . . 7  |-  -.  th
6564nex 1564 . . . . . 6  |-  -.  E. y th
661simp1bi 972 . . . . . . . . . 10  |-  ( ph  ->  R  FrSe  A )
672, 66bnj832 29063 . . . . . . . . 9  |-  ( ps 
->  R  FrSe  A )
684, 67bnj835 29065 . . . . . . . 8  |-  ( ch 
->  R  FrSe  A )
6948a1i 11 . . . . . . . 8  |-  ( ch 
->  D  C_  A )
704simp2bi 973 . . . . . . . . . 10  |-  ( ch 
->  x  e.  A
)
714simp3bi 974 . . . . . . . . . 10  |-  ( ch 
->  ( F `  x
)  =/=  ( H `
 x ) )
7230rabeq2i 2945 . . . . . . . . . 10  |-  ( x  e.  D  <->  ( x  e.  A  /\  ( F `  x )  =/=  ( H `  x
) ) )
7370, 71, 72sylanbrc 646 . . . . . . . . 9  |-  ( ch 
->  x  e.  D
)
74 ne0i 3626 . . . . . . . . 9  |-  ( x  e.  D  ->  D  =/=  (/) )
7573, 74syl 16 . . . . . . . 8  |-  ( ch 
->  D  =/=  (/) )
76 bnj69 29316 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  D  C_  A  /\  D  =/=  (/) )  ->  E. y  e.  D  A. z  e.  D  -.  z R y )
7768, 69, 75, 76syl3anc 1184 . . . . . . 7  |-  ( ch 
->  E. y  e.  D  A. z  e.  D  -.  z R y )
7877, 3bnj1209 29105 . . . . . 6  |-  ( ch 
->  E. y th )
7965, 78mto 169 . . . . 5  |-  -.  ch
8079nex 1564 . . . 4  |-  -.  E. x ch
812simprbi 451 . . . . . 6  |-  ( ps 
->  F  =/=  H
)
8211, 15, 81, 36bnj1542 29165 . . . . 5  |-  ( ps 
->  E. x  e.  A  ( F `  x )  =/=  ( H `  x ) )
835, 6, 7, 8, 1, 2bnj1525 29375 . . . . 5  |-  ( ps 
->  A. x ps )
8482, 4, 83bnj1521 29159 . . . 4  |-  ( ps 
->  E. x ch )
8580, 84mto 169 . . 3  |-  -.  ps
862, 85bnj1541 29164 . 2  |-  ( ph  ->  F  =  H )
871, 86sylbir 205 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 177    /\ wa 359    /\ w3a 936   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701    C_ wss 3312   (/)c0 3620   <.cop 3809   U.cuni 4007   class class class wbr 4204    |` cres 4872    Fn wfn 5441   ` cfv 5446    predc-bnj14 28989    FrSe w-bnj15 28993
This theorem is referenced by:  bnj1522  29378
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-reg 7552  ax-inf2 7588
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1o 6716  df-bnj17 28988  df-bnj14 28990  df-bnj13 28992  df-bnj15 28994  df-bnj18 28996  df-bnj19 28998
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