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Theorem bnj1501 29097
Description: Technical lemma for bnj1500 29098. 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
bnj1501.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1501.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1501.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1501.4  |-  F  = 
U. C
bnj1501.5  |-  ( ph  <->  ( R  FrSe  A  /\  x  e.  A )
)
bnj1501.6  |-  ( ps  <->  (
ph  /\  f  e.  C  /\  x  e.  dom  f ) )
bnj1501.7  |-  ( ch  <->  ( ps  /\  d  e.  B  /\  dom  f  =  d ) )
Assertion
Ref Expression
bnj1501  |-  ( R 
FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
Distinct variable groups:    A, d,
f, x    B, f    G, d, f, x    R, d, f, x    Y, d    ph, d, f
Allowed substitution hints:    ph( x)    ps( x, f, d)    ch( x, f, d)    B( x, d)    C( x, f, d)    F( x, f, d)    Y( x, f)

Proof of Theorem bnj1501
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 bnj1501.5 . 2  |-  ( ph  <->  ( R  FrSe  A  /\  x  e.  A )
)
21simprbi 450 . . . . . . . 8  |-  ( ph  ->  x  e.  A )
3 bnj1501.1 . . . . . . . . . . 11  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
4 bnj1501.2 . . . . . . . . . . 11  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
5 bnj1501.3 . . . . . . . . . . 11  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
6 bnj1501.4 . . . . . . . . . . 11  |-  F  = 
U. C
73, 4, 5, 6bnj60 29092 . . . . . . . . . 10  |-  ( R 
FrSe  A  ->  F  Fn  A )
8 fndm 5343 . . . . . . . . . 10  |-  ( F  Fn  A  ->  dom  F  =  A )
97, 8syl 15 . . . . . . . . 9  |-  ( R 
FrSe  A  ->  dom  F  =  A )
101, 9bnj832 28787 . . . . . . . 8  |-  ( ph  ->  dom  F  =  A )
112, 10eleqtrrd 2360 . . . . . . 7  |-  ( ph  ->  x  e.  dom  F
)
126dmeqi 4880 . . . . . . . 8  |-  dom  F  =  dom  U. C
135bnj1317 28854 . . . . . . . . 9  |-  ( w  e.  C  ->  A. f  w  e.  C )
1413bnj1400 28868 . . . . . . . 8  |-  dom  U. C  =  U_ f  e.  C  dom  f
1512, 14eqtri 2303 . . . . . . 7  |-  dom  F  =  U_ f  e.  C  dom  f
1611, 15syl6eleq 2373 . . . . . 6  |-  ( ph  ->  x  e.  U_ f  e.  C  dom  f )
1716bnj1405 28869 . . . . 5  |-  ( ph  ->  E. f  e.  C  x  e.  dom  f )
18 bnj1501.6 . . . . 5  |-  ( ps  <->  (
ph  /\  f  e.  C  /\  x  e.  dom  f ) )
1917, 18bnj1209 28829 . . . 4  |-  ( ph  ->  E. f ps )
205bnj1436 28872 . . . . . . . . . 10  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
2120bnj1299 28851 . . . . . . . . 9  |-  ( f  e.  C  ->  E. d  e.  B  f  Fn  d )
22 fndm 5343 . . . . . . . . 9  |-  ( f  Fn  d  ->  dom  f  =  d )
2321, 22bnj31 28745 . . . . . . . 8  |-  ( f  e.  C  ->  E. d  e.  B  dom  f  =  d )
2418, 23bnj836 28790 . . . . . . 7  |-  ( ps 
->  E. d  e.  B  dom  f  =  d
)
25 bnj1501.7 . . . . . . 7  |-  ( ch  <->  ( ps  /\  d  e.  B  /\  dom  f  =  d ) )
263, 4, 5, 6, 1, 18bnj1518 29094 . . . . . . 7  |-  ( ps 
->  A. d ps )
2724, 25, 26bnj1521 28883 . . . . . 6  |-  ( ps 
->  E. d ch )
287bnj930 28801 . . . . . . . . . . . 12  |-  ( R 
FrSe  A  ->  Fun  F
)
291, 28bnj832 28787 . . . . . . . . . . 11  |-  ( ph  ->  Fun  F )
3018, 29bnj835 28789 . . . . . . . . . 10  |-  ( ps 
->  Fun  F )
31 elssuni 3855 . . . . . . . . . . . 12  |-  ( f  e.  C  ->  f  C_ 
U. C )
3231, 6syl6sseqr 3225 . . . . . . . . . . 11  |-  ( f  e.  C  ->  f  C_  F )
3318, 32bnj836 28790 . . . . . . . . . 10  |-  ( ps 
->  f  C_  F )
3418simp3bi 972 . . . . . . . . . 10  |-  ( ps 
->  x  e.  dom  f )
3530, 33, 34bnj1502 28880 . . . . . . . . 9  |-  ( ps 
->  ( F `  x
)  =  ( f `
 x ) )
363, 4, 5bnj1514 29093 . . . . . . . . . . 11  |-  ( f  e.  C  ->  A. x  e.  dom  f ( f `
 x )  =  ( G `  Y
) )
3718, 36bnj836 28790 . . . . . . . . . 10  |-  ( ps 
->  A. x  e.  dom  f ( f `  x )  =  ( G `  Y ) )
3837, 34bnj1294 28850 . . . . . . . . 9  |-  ( ps 
->  ( f `  x
)  =  ( G `
 Y ) )
3935, 38eqtrd 2315 . . . . . . . 8  |-  ( ps 
->  ( F `  x
)  =  ( G `
 Y ) )
4025, 39bnj835 28789 . . . . . . 7  |-  ( ch 
->  ( F `  x
)  =  ( G `
 Y ) )
4125, 30bnj835 28789 . . . . . . . . . . 11  |-  ( ch 
->  Fun  F )
4225, 33bnj835 28789 . . . . . . . . . . 11  |-  ( ch 
->  f  C_  F )
433bnj1517 28882 . . . . . . . . . . . . . 14  |-  ( d  e.  B  ->  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
4425, 43bnj836 28790 . . . . . . . . . . . . 13  |-  ( ch 
->  A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d )
4525, 34bnj835 28789 . . . . . . . . . . . . . 14  |-  ( ch 
->  x  e.  dom  f )
4625simp3bi 972 . . . . . . . . . . . . . 14  |-  ( ch 
->  dom  f  =  d )
4745, 46eleqtrd 2359 . . . . . . . . . . . . 13  |-  ( ch 
->  x  e.  d
)
4844, 47bnj1294 28850 . . . . . . . . . . . 12  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  d )
4948, 46sseqtr4d 3215 . . . . . . . . . . 11  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  dom  f )
5041, 42, 49bnj1503 28881 . . . . . . . . . 10  |-  ( ch 
->  ( F  |`  pred (
x ,  A ,  R ) )  =  ( f  |`  pred (
x ,  A ,  R ) ) )
5150opeq2d 3803 . . . . . . . . 9  |-  ( ch 
->  <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >.  =  <. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
)
5251, 4syl6eqr 2333 . . . . . . . 8  |-  ( ch 
->  <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >.  =  Y )
5352fveq2d 5529 . . . . . . 7  |-  ( ch 
->  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
)  =  ( G `
 Y ) )
5440, 53eqtr4d 2318 . . . . . 6  |-  ( ch 
->  ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
5527, 54bnj593 28774 . . . . 5  |-  ( ps 
->  E. d ( F `
 x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
563, 4, 5, 6bnj1519 29095 . . . . 5  |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. )  ->  A. d
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
5755, 56bnj1397 28867 . . . 4  |-  ( ps 
->  ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
5819, 57bnj593 28774 . . 3  |-  ( ph  ->  E. f ( F `
 x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
593, 4, 5, 6bnj1520 29096 . . 3  |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. )  ->  A. f
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
6058, 59bnj1397 28867 . 2  |-  ( ph  ->  ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
611, 60bnj1459 28875 1  |-  ( R 
FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544    C_ wss 3152   <.cop 3643   U.cuni 3827   U_ciun 3905   dom cdm 4689    |` cres 4691   Fun wfun 5249    Fn wfn 5250   ` cfv 5255    predc-bnj14 28713    FrSe w-bnj15 28717
This theorem is referenced by:  bnj1500  29098
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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