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Theorem bnj1501 33202
Description: Technical lemma for bnj1500 33203. 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 33197 . . . . . . . . . 10  |-  ( R 
FrSe  A  ->  F  Fn  A )
8 fndm 5678 . . . . . . . . . 10  |-  ( F  Fn  A  ->  dom  F  =  A )
97, 8syl 16 . . . . . . . . 9  |-  ( R 
FrSe  A  ->  dom  F  =  A )
101, 9bnj832 32894 . . . . . . . 8  |-  ( ph  ->  dom  F  =  A )
112, 10eleqtrrd 2558 . . . . . . 7  |-  ( ph  ->  x  e.  dom  F
)
126dmeqi 5202 . . . . . . . 8  |-  dom  F  =  dom  U. C
135bnj1317 32959 . . . . . . . . 9  |-  ( w  e.  C  ->  A. f  w  e.  C )
1413bnj1400 32973 . . . . . . . 8  |-  dom  U. C  =  U_ f  e.  C  dom  f
1512, 14eqtri 2496 . . . . . . 7  |-  dom  F  =  U_ f  e.  C  dom  f
1611, 15syl6eleq 2565 . . . . . 6  |-  ( ph  ->  x  e.  U_ f  e.  C  dom  f )
1716bnj1405 32974 . . . . 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 32934 . . . 4  |-  ( ph  ->  E. f ps )
205bnj1436 32977 . . . . . . . . . 10  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
2120bnj1299 32956 . . . . . . . . 9  |-  ( f  e.  C  ->  E. d  e.  B  f  Fn  d )
22 fndm 5678 . . . . . . . . 9  |-  ( f  Fn  d  ->  dom  f  =  d )
2321, 22bnj31 32852 . . . . . . . 8  |-  ( f  e.  C  ->  E. d  e.  B  dom  f  =  d )
2418, 23bnj836 32897 . . . . . . 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 33199 . . . . . . 7  |-  ( ps 
->  A. d ps )
2724, 25, 26bnj1521 32988 . . . . . 6  |-  ( ps 
->  E. d ch )
287bnj930 32907 . . . . . . . . . . . 12  |-  ( R 
FrSe  A  ->  Fun  F
)
291, 28bnj832 32894 . . . . . . . . . . 11  |-  ( ph  ->  Fun  F )
3018, 29bnj835 32896 . . . . . . . . . 10  |-  ( ps 
->  Fun  F )
31 elssuni 4275 . . . . . . . . . . . 12  |-  ( f  e.  C  ->  f  C_ 
U. C )
3231, 6syl6sseqr 3551 . . . . . . . . . . 11  |-  ( f  e.  C  ->  f  C_  F )
3318, 32bnj836 32897 . . . . . . . . . 10  |-  ( ps 
->  f  C_  F )
3418simp3bi 1013 . . . . . . . . . 10  |-  ( ps 
->  x  e.  dom  f )
3530, 33, 34bnj1502 32985 . . . . . . . . 9  |-  ( ps 
->  ( F `  x
)  =  ( f `
 x ) )
363, 4, 5bnj1514 33198 . . . . . . . . . . 11  |-  ( f  e.  C  ->  A. x  e.  dom  f ( f `
 x )  =  ( G `  Y
) )
3718, 36bnj836 32897 . . . . . . . . . 10  |-  ( ps 
->  A. x  e.  dom  f ( f `  x )  =  ( G `  Y ) )
3837, 34bnj1294 32955 . . . . . . . . 9  |-  ( ps 
->  ( f `  x
)  =  ( G `
 Y ) )
3935, 38eqtrd 2508 . . . . . . . 8  |-  ( ps 
->  ( F `  x
)  =  ( G `
 Y ) )
4025, 39bnj835 32896 . . . . . . 7  |-  ( ch 
->  ( F `  x
)  =  ( G `
 Y ) )
4125, 30bnj835 32896 . . . . . . . . . . 11  |-  ( ch 
->  Fun  F )
4225, 33bnj835 32896 . . . . . . . . . . 11  |-  ( ch 
->  f  C_  F )
433bnj1517 32987 . . . . . . . . . . . . . 14  |-  ( d  e.  B  ->  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
4425, 43bnj836 32897 . . . . . . . . . . . . 13  |-  ( ch 
->  A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d )
4525, 34bnj835 32896 . . . . . . . . . . . . . 14  |-  ( ch 
->  x  e.  dom  f )
4625simp3bi 1013 . . . . . . . . . . . . . 14  |-  ( ch 
->  dom  f  =  d )
4745, 46eleqtrd 2557 . . . . . . . . . . . . 13  |-  ( ch 
->  x  e.  d
)
4844, 47bnj1294 32955 . . . . . . . . . . . 12  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  d )
4948, 46sseqtr4d 3541 . . . . . . . . . . 11  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  dom  f )
5041, 42, 49bnj1503 32986 . . . . . . . . . 10  |-  ( ch 
->  ( F  |`  pred (
x ,  A ,  R ) )  =  ( f  |`  pred (
x ,  A ,  R ) ) )
5150opeq2d 4220 . . . . . . . . 9  |-  ( ch 
->  <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >.  =  <. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
)
5251, 4syl6eqr 2526 . . . . . . . 8  |-  ( ch 
->  <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >.  =  Y )
5352fveq2d 5868 . . . . . . 7  |-  ( ch 
->  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
)  =  ( G `
 Y ) )
5440, 53eqtr4d 2511 . . . . . 6  |-  ( ch 
->  ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
5527, 54bnj593 32881 . . . . 5  |-  ( ps 
->  E. d ( F `
 x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
563, 4, 5, 6bnj1519 33200 . . . . 5  |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. )  ->  A. d
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
5755, 56bnj1397 32972 . . . 4  |-  ( ps 
->  ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
5819, 57bnj593 32881 . . 3  |-  ( ph  ->  E. f ( F `
 x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
593, 4, 5, 6bnj1520 33201 . . 3  |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. )  ->  A. f
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
6058, 59bnj1397 32972 . 2  |-  ( ph  ->  ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
611, 60bnj1459 32980 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 973    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814   E.wrex 2815    C_ wss 3476   <.cop 4033   U.cuni 4245   U_ciun 4325   dom cdm 4999    |` cres 5001   Fun wfun 5580    Fn wfn 5581   ` cfv 5586    predc-bnj14 32820    FrSe w-bnj15 32824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-reg 8014  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-om 6679  df-1o 7127  df-bnj17 32819  df-bnj14 32821  df-bnj13 32823  df-bnj15 32825  df-bnj18 32827  df-bnj19 32829
This theorem is referenced by:  bnj1500  33203
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