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Theorem bnj1501 31945
Description: Technical lemma for bnj1500 31946. 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 31940 . . . . . . . . . 10  |-  ( R 
FrSe  A  ->  F  Fn  A )
8 fndm 5505 . . . . . . . . . 10  |-  ( F  Fn  A  ->  dom  F  =  A )
97, 8syl 16 . . . . . . . . 9  |-  ( R 
FrSe  A  ->  dom  F  =  A )
101, 9bnj832 31637 . . . . . . . 8  |-  ( ph  ->  dom  F  =  A )
112, 10eleqtrrd 2515 . . . . . . 7  |-  ( ph  ->  x  e.  dom  F
)
126dmeqi 5036 . . . . . . . 8  |-  dom  F  =  dom  U. C
135bnj1317 31702 . . . . . . . . 9  |-  ( w  e.  C  ->  A. f  w  e.  C )
1413bnj1400 31716 . . . . . . . 8  |-  dom  U. C  =  U_ f  e.  C  dom  f
1512, 14eqtri 2458 . . . . . . 7  |-  dom  F  =  U_ f  e.  C  dom  f
1611, 15syl6eleq 2528 . . . . . 6  |-  ( ph  ->  x  e.  U_ f  e.  C  dom  f )
1716bnj1405 31717 . . . . 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 31677 . . . 4  |-  ( ph  ->  E. f ps )
205bnj1436 31720 . . . . . . . . . 10  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
2120bnj1299 31699 . . . . . . . . 9  |-  ( f  e.  C  ->  E. d  e.  B  f  Fn  d )
22 fndm 5505 . . . . . . . . 9  |-  ( f  Fn  d  ->  dom  f  =  d )
2321, 22bnj31 31595 . . . . . . . 8  |-  ( f  e.  C  ->  E. d  e.  B  dom  f  =  d )
2418, 23bnj836 31640 . . . . . . 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 31942 . . . . . . 7  |-  ( ps 
->  A. d ps )
2724, 25, 26bnj1521 31731 . . . . . 6  |-  ( ps 
->  E. d ch )
287bnj930 31650 . . . . . . . . . . . 12  |-  ( R 
FrSe  A  ->  Fun  F
)
291, 28bnj832 31637 . . . . . . . . . . 11  |-  ( ph  ->  Fun  F )
3018, 29bnj835 31639 . . . . . . . . . 10  |-  ( ps 
->  Fun  F )
31 elssuni 4116 . . . . . . . . . . . 12  |-  ( f  e.  C  ->  f  C_ 
U. C )
3231, 6syl6sseqr 3398 . . . . . . . . . . 11  |-  ( f  e.  C  ->  f  C_  F )
3318, 32bnj836 31640 . . . . . . . . . 10  |-  ( ps 
->  f  C_  F )
3418simp3bi 1005 . . . . . . . . . 10  |-  ( ps 
->  x  e.  dom  f )
3530, 33, 34bnj1502 31728 . . . . . . . . 9  |-  ( ps 
->  ( F `  x
)  =  ( f `
 x ) )
363, 4, 5bnj1514 31941 . . . . . . . . . . 11  |-  ( f  e.  C  ->  A. x  e.  dom  f ( f `
 x )  =  ( G `  Y
) )
3718, 36bnj836 31640 . . . . . . . . . 10  |-  ( ps 
->  A. x  e.  dom  f ( f `  x )  =  ( G `  Y ) )
3837, 34bnj1294 31698 . . . . . . . . 9  |-  ( ps 
->  ( f `  x
)  =  ( G `
 Y ) )
3935, 38eqtrd 2470 . . . . . . . 8  |-  ( ps 
->  ( F `  x
)  =  ( G `
 Y ) )
4025, 39bnj835 31639 . . . . . . 7  |-  ( ch 
->  ( F `  x
)  =  ( G `
 Y ) )
4125, 30bnj835 31639 . . . . . . . . . . 11  |-  ( ch 
->  Fun  F )
4225, 33bnj835 31639 . . . . . . . . . . 11  |-  ( ch 
->  f  C_  F )
433bnj1517 31730 . . . . . . . . . . . . . 14  |-  ( d  e.  B  ->  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
4425, 43bnj836 31640 . . . . . . . . . . . . 13  |-  ( ch 
->  A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d )
4525, 34bnj835 31639 . . . . . . . . . . . . . 14  |-  ( ch 
->  x  e.  dom  f )
4625simp3bi 1005 . . . . . . . . . . . . . 14  |-  ( ch 
->  dom  f  =  d )
4745, 46eleqtrd 2514 . . . . . . . . . . . . 13  |-  ( ch 
->  x  e.  d
)
4844, 47bnj1294 31698 . . . . . . . . . . . 12  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  d )
4948, 46sseqtr4d 3388 . . . . . . . . . . 11  |-  ( ch 
->  pred ( x ,  A ,  R ) 
C_  dom  f )
5041, 42, 49bnj1503 31729 . . . . . . . . . 10  |-  ( ch 
->  ( F  |`  pred (
x ,  A ,  R ) )  =  ( f  |`  pred (
x ,  A ,  R ) ) )
5150opeq2d 4061 . . . . . . . . 9  |-  ( ch 
->  <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >.  =  <. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
)
5251, 4syl6eqr 2488 . . . . . . . 8  |-  ( ch 
->  <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >.  =  Y )
5352fveq2d 5690 . . . . . . 7  |-  ( ch 
->  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
)  =  ( G `
 Y ) )
5440, 53eqtr4d 2473 . . . . . 6  |-  ( ch 
->  ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
5527, 54bnj593 31624 . . . . 5  |-  ( ps 
->  E. d ( F `
 x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
563, 4, 5, 6bnj1519 31943 . . . . 5  |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. )  ->  A. d
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
5755, 56bnj1397 31715 . . . 4  |-  ( ps 
->  ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
5819, 57bnj593 31624 . . 3  |-  ( ph  ->  E. f ( F `
 x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
593, 4, 5, 6bnj1520 31944 . . 3  |-  ( ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >. )  ->  A. f
( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
6058, 59bnj1397 31715 . 2  |-  ( ph  ->  ( F `  x
)  =  ( G `
 <. x ,  ( F  |`  pred ( x ,  A ,  R
) ) >. )
)
611, 60bnj1459 31723 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 1369    e. wcel 1756   {cab 2424   A.wral 2710   E.wrex 2711    C_ wss 3323   <.cop 3878   U.cuni 4086   U_ciun 4166   dom cdm 4835    |` cres 4837   Fun wfun 5407    Fn wfn 5408   ` cfv 5413    predc-bnj14 31563    FrSe w-bnj15 31567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-reg 7799  ax-inf2 7839
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-om 6472  df-1o 6912  df-bnj17 31562  df-bnj14 31564  df-bnj13 31566  df-bnj15 31568  df-bnj18 31570  df-bnj19 31572
This theorem is referenced by:  bnj1500  31946
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