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Theorem bnj1501 32360
Description: Technical lemma for bnj1500 32361. 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 464 . . . . . . . 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 32355 . . . . . . . . . 10  |-  ( R 
FrSe  A  ->  F  Fn  A )
8 fndm 5610 . . . . . . . . . 10  |-  ( F  Fn  A  ->  dom  F  =  A )
97, 8syl 16 . . . . . . . . 9  |-  ( R 
FrSe  A  ->  dom  F  =  A )
101, 9bnj832 32052 . . . . . . . 8  |-  ( ph  ->  dom  F  =  A )
112, 10eleqtrrd 2542 . . . . . . 7  |-  ( ph  ->  x  e.  dom  F
)
126dmeqi 5141 . . . . . . . 8  |-  dom  F  =  dom  U. C
135bnj1317 32117 . . . . . . . . 9  |-  ( w  e.  C  ->  A. f  w  e.  C )
1413bnj1400 32131 . . . . . . . 8  |-  dom  U. C  =  U_ f  e.  C  dom  f
1512, 14eqtri 2480 . . . . . . 7  |-  dom  F  =  U_ f  e.  C  dom  f
1611, 15syl6eleq 2549 . . . . . 6  |-  ( ph  ->  x  e.  U_ f  e.  C  dom  f )
1716bnj1405 32132 . . . . 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 32092 . . . 4  |-  ( ph  ->  E. f ps )
205bnj1436 32135 . . . . . . . . . 10  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
2120bnj1299 32114 . . . . . . . . 9  |-  ( f  e.  C  ->  E. d  e.  B  f  Fn  d )
22 fndm 5610 . . . . . . . . 9  |-  ( f  Fn  d  ->  dom  f  =  d )
2321, 22bnj31 32010 . . . . . . . 8  |-  ( f  e.  C  ->  E. d  e.  B  dom  f  =  d )
2418, 23bnj836 32055 . . . . . . 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 32357 . . . . . . 7  |-  ( ps 
->  A. d ps )
2724, 25, 26bnj1521 32146 . . . . . 6  |-  ( ps 
->  E. d ch )
287bnj930 32065 . . . . . . . . . . . 12  |-  ( R 
FrSe  A  ->  Fun  F
)
291, 28bnj832 32052 . . . . . . . . . . 11  |-  ( ph  ->  Fun  F )
3018, 29bnj835 32054 . . . . . . . . . 10  |-  ( ps 
->  Fun  F )
31 elssuni 4221 . . . . . . . . . . . 12  |-  ( f  e.  C  ->  f  C_ 
U. C )
3231, 6syl6sseqr 3503 . . . . . . . . . . 11  |-  ( f  e.  C  ->  f  C_  F )
3318, 32bnj836 32055 . . . . . . . . . 10  |-  ( ps 
->  f  C_  F )
3418simp3bi 1005 . . . . . . . . . 10  |-  ( ps 
->  x  e.  dom  f )
3530, 33, 34bnj1502 32143 . . . . . . . . 9  |-  ( ps 
->  ( F `  x
)  =  ( f `
 x ) )
363, 4, 5bnj1514 32356 . . . . . . . . . . 11  |-  ( f  e.  C  ->  A. x  e.  dom  f ( f `
 x )  =  ( G `  Y
) )
3718, 36bnj836 32055 . . . . . . . . . 10  |-  ( ps 
->  A. x  e.  dom  f ( f `  x )  =  ( G `  Y ) )
3837, 34bnj1294 32113 . . . . . . . . 9  |-  ( ps 
->  ( f `  x
)  =  ( G `
 Y ) )
3935, 38eqtrd 2492 . . . . . . . 8  |-  ( ps 
->  ( F `  x
)  =  ( G `
 Y ) )
4025, 39bnj835 32054 . . . . . . 7  |-  ( ch 
->  ( F `  x
)  =  ( G `
 Y ) )
4125, 30bnj835 32054 . . . . . . . . . . 11  |-  ( ch 
->  Fun  F )
4225, 33bnj835 32054 . . . . . . . . . . 11  |-  ( ch 
->  f  C_  F )
433bnj1517 32145 . . . . . . . . . . . . . 14  |-  ( d  e.  B  ->  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
4425, 43bnj836 32055 . . . . . . . . . . . . 13  |-  ( ch 
->  A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d )
4525, 34bnj835 32054 . . . . . . . . . . . . . 14  |-  ( ch 
->  x  e.  dom  f )
4625simp3bi 1005 . . . . . . . . . . . . . 14  |-  ( ch 
->  dom  f  =  d )
4745, 46eleqtrd 2541 . . . . . . . . . . . . 13  |-  ( ch 
->  x  e.  d
)
4844, 47bnj1294 32113 . . . . . . . . . . . 12  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  d )
4948, 46sseqtr4d 3493 . . . . . . . . . . 11  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  dom  f )
5041, 42, 49bnj1503 32144 . . . . . . . . . 10  |-  ( ch 
->  ( F  |`  pred (
x ,  A ,  R ) )  =  ( f  |`  pred (
x ,  A ,  R ) ) )
5150opeq2d 4166 . . . . . . . . 9  |-  ( ch 
->  <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >.  =  <. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
)
5251, 4syl6eqr 2510 . . . . . . . 8  |-  ( ch 
->  <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >.  =  Y )
5352fveq2d 5795 . . . . . . 7  |-  ( ch 
->  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
)  =  ( G `
 Y ) )
5440, 53eqtr4d 2495 . . . . . 6  |-  ( ch 
->  ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
5527, 54bnj593 32039 . . . . 5  |-  ( ps 
->  E. d ( F `
 x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
563, 4, 5, 6bnj1519 32358 . . . . 5  |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. )  ->  A. d
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
5755, 56bnj1397 32130 . . . 4  |-  ( ps 
->  ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
5819, 57bnj593 32039 . . 3  |-  ( ph  ->  E. f ( F `
 x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
593, 4, 5, 6bnj1520 32359 . . 3  |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. )  ->  A. f
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
6058, 59bnj1397 32130 . 2  |-  ( ph  ->  ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
611, 60bnj1459 32138 1  |-  ( R 
FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {cab 2436   A.wral 2795   E.wrex 2796    C_ wss 3428   <.cop 3983   U.cuni 4191   U_ciun 4271   dom cdm 4940    |` cres 4942   Fun wfun 5512    Fn wfn 5513   ` cfv 5518    predc-bnj14 31978    FrSe w-bnj15 31982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-reg 7910  ax-inf2 7950
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-om 6579  df-1o 7022  df-bnj17 31977  df-bnj14 31979  df-bnj13 31981  df-bnj15 31983  df-bnj18 31985  df-bnj19 31987
This theorem is referenced by:  bnj1500  32361
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