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Theorem bnj1384 29062
Description: Technical lemma for bnj60 29092. 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
bnj1384.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1384.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1384.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1384.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1384.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1384.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1384.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1384.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1384.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1384.10  |-  P  = 
U. H
Assertion
Ref Expression
bnj1384  |-  ( R 
FrSe  A  ->  Fun  P
)
Distinct variable groups:    A, d,
f, x    B, f    y, C    G, d, f    R, d, f, x    y, f, x
Allowed substitution hints:    ps( x, y, f, d)    ch( x, y, f, d)    ta( x, y, f, d)    A( y)    B( x, y, d)    C( x, f, d)    D( x, y, f, d)    P( x, y, f, d)    R( y)    G( x, y)    H( x, y, f, d)    Y( x, y, f, d)    ta'( x, y, f, d)

Proof of Theorem bnj1384
Dummy variables  z 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj1384.1 . . . . 5  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
2 bnj1384.2 . . . . 5  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
3 bnj1384.3 . . . . 5  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
4 bnj1384.4 . . . . 5  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
5 bnj1384.5 . . . . 5  |-  D  =  { x  e.  A  |  -.  E. f ta }
6 bnj1384.6 . . . . 5  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
7 bnj1384.7 . . . . 5  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
8 bnj1384.8 . . . . 5  |-  ( ta'  <->  [. y  /  x ]. ta )
9 bnj1384.9 . . . . 5  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
10 bnj1384.10 . . . . 5  |-  P  = 
U. H
111, 2, 3, 4, 8bnj1373 29060 . . . . 5  |-  ( ta'  <->  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11bnj1371 29059 . . . 4  |-  ( f  e.  H  ->  Fun  f )
1312rgen 2608 . . 3  |-  A. f  e.  H  Fun  f
14 id 19 . . . . . 6  |-  ( R 
FrSe  A  ->  R  FrSe  A )
151, 2, 3, 4, 5, 6, 7, 8, 9bnj1374 29061 . . . . . 6  |-  ( f  e.  H  ->  f  e.  C )
16 nfab1 2421 . . . . . . . . . 10  |-  F/_ f { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
179, 16nfcxfr 2416 . . . . . . . . 9  |-  F/_ f H
1817nfcri 2413 . . . . . . . 8  |-  F/ f  g  e.  H
19 nfab1 2421 . . . . . . . . . 10  |-  F/_ f { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
203, 19nfcxfr 2416 . . . . . . . . 9  |-  F/_ f C
2120nfcri 2413 . . . . . . . 8  |-  F/ f  g  e.  C
2218, 21nfim 1769 . . . . . . 7  |-  F/ f ( g  e.  H  ->  g  e.  C )
23 eleq1 2343 . . . . . . . 8  |-  ( f  =  g  ->  (
f  e.  H  <->  g  e.  H ) )
24 eleq1 2343 . . . . . . . 8  |-  ( f  =  g  ->  (
f  e.  C  <->  g  e.  C ) )
2523, 24imbi12d 311 . . . . . . 7  |-  ( f  =  g  ->  (
( f  e.  H  ->  f  e.  C )  <-> 
( g  e.  H  ->  g  e.  C ) ) )
2622, 25, 15chvar 1926 . . . . . 6  |-  ( g  e.  H  ->  g  e.  C )
27 eqid 2283 . . . . . . 7  |-  ( dom  f  i^i  dom  g
)  =  ( dom  f  i^i  dom  g
)
281, 2, 3, 27bnj1326 29056 . . . . . 6  |-  ( ( R  FrSe  A  /\  f  e.  C  /\  g  e.  C )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  ( dom  f  i^i  dom  g )
) )
2914, 15, 26, 28syl3an 1224 . . . . 5  |-  ( ( R  FrSe  A  /\  f  e.  H  /\  g  e.  H )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  ( dom  f  i^i  dom  g )
) )
30293expib 1154 . . . 4  |-  ( R 
FrSe  A  ->  ( ( f  e.  H  /\  g  e.  H )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  ( dom  f  i^i  dom  g )
) ) )
3130ralrimivv 2634 . . 3  |-  ( R 
FrSe  A  ->  A. f  e.  H  A. g  e.  H  ( f  |`  ( dom  f  i^i 
dom  g ) )  =  ( g  |`  ( dom  f  i^i  dom  g ) ) )
32 biid 227 . . . 4  |-  ( A. f  e.  H  Fun  f 
<-> 
A. f  e.  H  Fun  f )
33 biid 227 . . . 4  |-  ( ( A. f  e.  H  Fun  f  /\  A. f  e.  H  A. g  e.  H  ( f  |`  ( dom  f  i^i 
dom  g ) )  =  ( g  |`  ( dom  f  i^i  dom  g ) ) )  <-> 
( A. f  e.  H  Fun  f  /\  A. f  e.  H  A. g  e.  H  (
f  |`  ( dom  f  i^i  dom  g ) )  =  ( g  |`  ( dom  f  i^i  dom  g ) ) ) )
349bnj1317 28854 . . . 4  |-  ( z  e.  H  ->  A. f 
z  e.  H )
3532, 27, 33, 34bnj1386 28866 . . 3  |-  ( ( A. f  e.  H  Fun  f  /\  A. f  e.  H  A. g  e.  H  ( f  |`  ( dom  f  i^i 
dom  g ) )  =  ( g  |`  ( dom  f  i^i  dom  g ) ) )  ->  Fun  U. H )
3613, 31, 35sylancr 644 . 2  |-  ( R 
FrSe  A  ->  Fun  U. H )
3710funeqi 5275 . 2  |-  ( Fun 
P  <->  Fun  U. H )
3836, 37sylibr 203 1  |-  ( R 
FrSe  A  ->  Fun  P
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   [.wsbc 2991    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   U.cuni 3827   class class class wbr 4023   dom cdm 4689    |` cres 4691   Fun wfun 5249    Fn wfn 5250   ` cfv 5255    predc-bnj14 28713    FrSe w-bnj15 28717    trClc-bnj18 28719
This theorem is referenced by:  bnj1312  29088
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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