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Theorem bnj1523 29417
Description: Technical lemma for bnj1522 29418. 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 29414 . . . . . . . . . . . . . . 15  |-  ( R 
FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
101, 9bnj835 29105 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. ) )
112, 10bnj832 29103 . . . . . . . . . . . . 13  |-  ( ps 
->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. ) )
124, 11bnj835 29105 . . . . . . . . . . . 12  |-  ( ch 
->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. ) )
135bnj1309 29368 . . . . . . . . . . . . . . . . 17  |-  ( w  e.  B  ->  A. x  w  e.  B )
147, 13bnj1307 29369 . . . . . . . . . . . . . . . 16  |-  ( w  e.  C  ->  A. x  w  e.  C )
1514nfcii 2423 . . . . . . . . . . . . . . 15  |-  F/_ x C
1615nfuni 3849 . . . . . . . . . . . . . 14  |-  F/_ x U. C
178, 16nfcxfr 2429 . . . . . . . . . . . . 13  |-  F/_ x F
1817nfcrii 2425 . . . . . . . . . . . 12  |-  ( w  e.  F  ->  A. x  w  e.  F )
1912, 18bnj1529 29416 . . . . . . . . . . 11  |-  ( ch 
->  A. y  e.  A  ( F `  y )  =  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >. ) )
203, 19bnj835 29105 . . . . . . . . . 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 29059 . . . . . . . . . . 11  |-  D  C_  A
233simp2bi 971 . . . . . . . . . . 11  |-  ( th 
->  y  e.  D
)
2422, 23bnj1213 29147 . . . . . . . . . 10  |-  ( th 
->  y  e.  A
)
2520, 24bnj1294 29166 . . . . . . . . 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 29103 . . . . . . . . . . . . . 14  |-  ( ps 
->  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )
284, 27bnj835 29105 . . . . . . . . . . . . 13  |-  ( ch 
->  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )
29 ax-17 1606 . . . . . . . . . . . . 13  |-  ( v  e.  H  ->  A. x  v  e.  H )
3028, 29bnj1529 29416 . . . . . . . . . . . 12  |-  ( ch 
->  A. y  e.  A  ( H `  y )  =  ( G `  <. y ,  ( H  |`  pred ( y ,  A ,  R ) ) >. ) )
313, 30bnj835 29105 . . . . . . . . . . 11  |-  ( th 
->  A. y  e.  A  ( H `  y )  =  ( G `  <. y ,  ( H  |`  pred ( y ,  A ,  R ) ) >. ) )
3231, 24bnj1294 29166 . . . . . . . . . 10  |-  ( th 
->  ( H `  y
)  =  ( G `
 <. y ,  ( H  |`  pred ( y ,  A ,  R
) ) >. )
)
335, 6, 7, 8bnj60 29408 . . . . . . . . . . . . . . . . 17  |-  ( R 
FrSe  A  ->  F  Fn  A )
341, 33bnj835 29105 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  F  Fn  A )
352, 34bnj832 29103 . . . . . . . . . . . . . . 15  |-  ( ps 
->  F  Fn  A
)
364, 35bnj835 29105 . . . . . . . . . . . . . 14  |-  ( ch 
->  F  Fn  A
)
373, 36bnj835 29105 . . . . . . . . . . . . 13  |-  ( th 
->  F  Fn  A
)
381simp2bi 971 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  H  Fn  A )
392, 38bnj832 29103 . . . . . . . . . . . . . . 15  |-  ( ps 
->  H  Fn  A
)
404, 39bnj835 29105 . . . . . . . . . . . . . 14  |-  ( ch 
->  H  Fn  A
)
413, 40bnj835 29105 . . . . . . . . . . . . 13  |-  ( th 
->  H  Fn  A
)
42 bnj213 29230 . . . . . . . . . . . . . 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 29146 . . . . . . . . . . . . . . . . 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 1556 . . . . . . . . . . . . . . . . 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 29386 . . . . . . . . . . . . . . . . . 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 1549 . . . . . . . . . . . . . . . 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 29137 . . . . . . . . . . . . . 14  |-  ( th 
->  A. z  e.  pred  ( y ,  A ,  R )  -.  z  e.  D )
5421, 18bnj1534 29201 . . . . . . . . . . . . . 14  |-  D  =  { z  e.  A  |  ( F `  z )  =/=  ( H `  z ) }
5553, 42, 54bnj1533 29200 . . . . . . . . . . . . 13  |-  ( th 
->  A. z  e.  pred  ( y ,  A ,  R ) ( F `
 z )  =  ( H `  z
) )
5637, 41, 43, 55bnj1536 29202 . . . . . . . . . . . 12  |-  ( th 
->  ( F  |`  pred (
y ,  A ,  R ) )  =  ( H  |`  pred (
y ,  A ,  R ) ) )
5756opeq2d 3819 . . . . . . . . . . 11  |-  ( th 
->  <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >.  =  <. y ,  ( H  |`  pred ( y ,  A ,  R ) ) >.
)
5857fveq2d 5545 . . . . . . . . . 10  |-  ( th 
->  ( G `  <. y ,  ( F  |`  pred ( y ,  A ,  R ) ) >.
)  =  ( G `
 <. y ,  ( H  |`  pred ( y ,  A ,  R
) ) >. )
)
5932, 58eqtr4d 2331 . . . . . . . . 9  |-  ( th 
->  ( H `  y
)  =  ( G `
 <. y ,  ( F  |`  pred ( y ,  A ,  R
) ) >. )
)
6025, 59eqtr4d 2331 . . . . . . . 8  |-  ( th 
->  ( F `  y
)  =  ( H `
 y ) )
6121, 18bnj1534 29201 . . . . . . . . . . 11  |-  D  =  { y  e.  A  |  ( F `  y )  =/=  ( H `  y ) }
6261bnj1538 29203 . . . . . . . . . 10  |-  ( y  e.  D  ->  ( F `  y )  =/=  ( H `  y
) )
633, 62bnj836 29106 . . . . . . . . 9  |-  ( th 
->  ( F `  y
)  =/=  ( H `
 y ) )
6463neneqd 2475 . . . . . . . 8  |-  ( th 
->  -.  ( F `  y )  =  ( H `  y ) )
6560, 64pm2.65i 165 . . . . . . 7  |-  -.  th
6665nex 1545 . . . . . 6  |-  -.  E. y th
671simp1bi 970 . . . . . . . . . 10  |-  ( ph  ->  R  FrSe  A )
682, 67bnj832 29103 . . . . . . . . 9  |-  ( ps 
->  R  FrSe  A )
694, 68bnj835 29105 . . . . . . . 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 2798 . . . . . . . . . 10  |-  ( x  e.  D  <->  ( x  e.  A  /\  ( F `  x )  =/=  ( H `  x
) ) )
7471, 72, 73sylanbrc 645 . . . . . . . . 9  |-  ( ch 
->  x  e.  D
)
75 ne0i 3474 . . . . . . . . 9  |-  ( x  e.  D  ->  D  =/=  (/) )
7674, 75syl 15 . . . . . . . 8  |-  ( ch 
->  D  =/=  (/) )
77 bnj69 29356 . . . . . . . 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 29145 . . . . . 6  |-  ( ch 
->  E. y th )
8066, 79mto 167 . . . . 5  |-  -.  ch
8180nex 1545 . . . 4  |-  -.  E. x ch
822simprbi 450 . . . . . 6  |-  ( ps 
->  F  =/=  H
)
8335, 39, 82, 18bnj1542 29205 . . . . 5  |-  ( ps 
->  E. x  e.  A  ( F `  x )  =/=  ( H `  x ) )
845, 6, 7, 8, 1, 2bnj1525 29415 . . . . 5  |-  ( ps 
->  A. x ps )
8583, 4, 84bnj1521 29199 . . . 4  |-  ( ps 
->  E. x ch )
8681, 85mto 167 . . 3  |-  -.  ps
872, 86bnj1541 29204 . 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 1530   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    C_ wss 3165   (/)c0 3468   <.cop 3656   U.cuni 3843   class class class wbr 4039    |` cres 4707    Fn wfn 5266   ` cfv 5271    predc-bnj14 29029    FrSe w-bnj15 29033
This theorem is referenced by:  bnj1522  29418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-bnj17 29028  df-bnj14 29030  df-bnj13 29032  df-bnj15 29034  df-bnj18 29036  df-bnj19 29038
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