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Theorem bnj1384 32119
Description: Technical lemma for bnj60 32149. 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 32117 . . . . 5  |-  ( ta'  <->  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11bnj1371 32116 . . . 4  |-  ( f  e.  H  ->  Fun  f )
1312rgen 2802 . . 3  |-  A. f  e.  H  Fun  f
14 id 22 . . . . . 6  |-  ( R 
FrSe  A  ->  R  FrSe  A )
151, 2, 3, 4, 5, 6, 7, 8, 9bnj1374 32118 . . . . . 6  |-  ( f  e.  H  ->  f  e.  C )
16 nfab1 2591 . . . . . . . . . 10  |-  F/_ f { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
179, 16nfcxfr 2587 . . . . . . . . 9  |-  F/_ f H
1817nfcri 2582 . . . . . . . 8  |-  F/ f  g  e.  H
19 nfab1 2591 . . . . . . . . . 10  |-  F/_ f { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
203, 19nfcxfr 2587 . . . . . . . . 9  |-  F/_ f C
2120nfcri 2582 . . . . . . . 8  |-  F/ f  g  e.  C
2218, 21nfim 1853 . . . . . . 7  |-  F/ f ( g  e.  H  ->  g  e.  C )
23 eleq1 2503 . . . . . . . 8  |-  ( f  =  g  ->  (
f  e.  H  <->  g  e.  H ) )
24 eleq1 2503 . . . . . . . 8  |-  ( f  =  g  ->  (
f  e.  C  <->  g  e.  C ) )
2523, 24imbi12d 320 . . . . . . 7  |-  ( f  =  g  ->  (
( f  e.  H  ->  f  e.  C )  <-> 
( g  e.  H  ->  g  e.  C ) ) )
2622, 25, 15chvar 1957 . . . . . 6  |-  ( g  e.  H  ->  g  e.  C )
27 eqid 2443 . . . . . . 7  |-  ( dom  f  i^i  dom  g
)  =  ( dom  f  i^i  dom  g
)
281, 2, 3, 27bnj1326 32113 . . . . . 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 1260 . . . . 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 1190 . . . 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 2828 . . 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 236 . . . 4  |-  ( A. f  e.  H  Fun  f 
<-> 
A. f  e.  H  Fun  f )
33 biid 236 . . . 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 31911 . . . 4  |-  ( z  e.  H  ->  A. f 
z  e.  H )
3532, 27, 33, 34bnj1386 31923 . . 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 663 . 2  |-  ( R 
FrSe  A  ->  Fun  U. H )
3710funeqi 5459 . 2  |-  ( Fun 
P  <->  Fun  U. H )
3836, 37sylibr 212 1  |-  ( R 
FrSe  A  ->  Fun  P
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429    =/= wne 2620   A.wral 2736   E.wrex 2737   {crab 2740   [.wsbc 3207    u. cun 3347    i^i cin 3348    C_ wss 3349   (/)c0 3658   {csn 3898   <.cop 3904   U.cuni 4112   class class class wbr 4313   dom cdm 4861    |` cres 4863   Fun wfun 5433    Fn wfn 5434   ` cfv 5439    predc-bnj14 31772    FrSe w-bnj15 31776    trClc-bnj18 31778
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 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-reg 7828  ax-inf2 7868
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-om 6498  df-1o 6941  df-bnj17 31771  df-bnj14 31773  df-bnj13 31775  df-bnj15 31777  df-bnj18 31779  df-bnj19 31781
This theorem is referenced by:  bnj1312  32145
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