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Theorem bnj1421 34241
Description: Technical lemma for bnj60 34261. 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
bnj1421.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1421.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1421.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1421.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1421.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1421.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1421.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1421.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1421.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1421.10  |-  P  = 
U. H
bnj1421.11  |-  Z  = 
<. x ,  ( P  |`  pred ( x ,  A ,  R ) ) >.
bnj1421.12  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
bnj1421.13  |-  ( ch 
->  Fun  P )
bnj1421.14  |-  ( ch 
->  dom  Q  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
bnj1421.15  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
Assertion
Ref Expression
bnj1421  |-  ( ch 
->  Fun  Q )
Distinct variable groups:    x, A    x, R
Allowed substitution hints:    ps( x, y, f, d)    ch( x, y, f, d)    ta( x, y, f, d)    A( y, f, d)    B( x, y, f, d)    C( x, y, f, d)    D( x, y, f, d)    P( x, y, f, d)    Q( x, y, f, d)    R( y, f, d)    G( x, y, f, d)    H( x, y, f, d)    Y( x, y, f, d)    Z( x, y, f, d)    ta'( x, y, f, d)

Proof of Theorem bnj1421
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 bnj1421.13 . . . 4  |-  ( ch 
->  Fun  P )
2 vex 3112 . . . . 5  |-  x  e. 
_V
3 fvex 5882 . . . . 5  |-  ( G `
 Z )  e. 
_V
42, 3funsn 5642 . . . 4  |-  Fun  { <. x ,  ( G `
 Z ) >. }
51, 4jctir 538 . . 3  |-  ( ch 
->  ( Fun  P  /\  Fun  { <. x ,  ( G `  Z )
>. } ) )
6 bnj1421.15 . . . . 5  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
73dmsnop 5488 . . . . . 6  |-  dom  { <. x ,  ( G `
 Z ) >. }  =  { x }
87a1i 11 . . . . 5  |-  ( ch 
->  dom  { <. x ,  ( G `  Z ) >. }  =  { x } )
96, 8ineq12d 3697 . . . 4  |-  ( ch 
->  ( dom  P  i^i  dom 
{ <. x ,  ( G `  Z )
>. } )  =  ( 
trCl ( x ,  A ,  R )  i^i  { x }
) )
10 bnj1421.7 . . . . . . 7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
11 bnj1421.6 . . . . . . . 8  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
1211simplbi 460 . . . . . . 7  |-  ( ps 
->  R  FrSe  A )
1310, 12bnj835 33960 . . . . . 6  |-  ( ch 
->  R  FrSe  A )
14 biid 236 . . . . . . . 8  |-  ( R 
FrSe  A  <->  R  FrSe  A )
15 biid 236 . . . . . . . 8  |-  ( -.  x  e.  trCl (
x ,  A ,  R )  <->  -.  x  e.  trCl ( x ,  A ,  R ) )
16 biid 236 . . . . . . . 8  |-  ( A. z  e.  A  (
z R x  ->  [. z  /  x ].  -.  x  e.  trCl ( x ,  A ,  R ) )  <->  A. z  e.  A  ( z R x  ->  [. z  /  x ].  -.  x  e.  trCl ( x ,  A ,  R ) ) )
17 biid 236 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  x  e.  A  /\  A. z  e.  A  ( z R x  ->  [. z  /  x ].  -.  x  e.  trCl ( x ,  A ,  R ) ) )  <-> 
( R  FrSe  A  /\  x  e.  A  /\  A. z  e.  A  ( z R x  ->  [. z  /  x ].  -.  x  e.  trCl ( x ,  A ,  R ) ) ) )
18 eqid 2457 . . . . . . . 8  |-  (  pred ( x ,  A ,  R )  u.  U_ z  e.  pred  ( x ,  A ,  R
)  trCl ( z ,  A ,  R ) )  =  (  pred ( x ,  A ,  R )  u.  U_ z  e.  pred  ( x ,  A ,  R
)  trCl ( z ,  A ,  R ) )
1914, 15, 16, 17, 18bnj1417 34240 . . . . . . 7  |-  ( R 
FrSe  A  ->  A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R ) )
20 disjsn 4092 . . . . . . . 8  |-  ( ( 
trCl ( x ,  A ,  R )  i^i  { x }
)  =  (/)  <->  -.  x  e.  trCl ( x ,  A ,  R ) )
2120ralbii 2888 . . . . . . 7  |-  ( A. x  e.  A  (  trCl ( x ,  A ,  R )  i^i  {
x } )  =  (/) 
<-> 
A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R ) )
2219, 21sylibr 212 . . . . . 6  |-  ( R 
FrSe  A  ->  A. x  e.  A  (  trCl ( x ,  A ,  R )  i^i  {
x } )  =  (/) )
2313, 22syl 16 . . . . 5  |-  ( ch 
->  A. x  e.  A  (  trCl ( x ,  A ,  R )  i^i  { x }
)  =  (/) )
24 bnj1421.5 . . . . . 6  |-  D  =  { x  e.  A  |  -.  E. f ta }
2524, 10bnj1212 34001 . . . . 5  |-  ( ch 
->  x  e.  A
)
2623, 25bnj1294 34019 . . . 4  |-  ( ch 
->  (  trCl ( x ,  A ,  R
)  i^i  { x } )  =  (/) )
279, 26eqtrd 2498 . . 3  |-  ( ch 
->  ( dom  P  i^i  dom 
{ <. x ,  ( G `  Z )
>. } )  =  (/) )
28 funun 5636 . . 3  |-  ( ( ( Fun  P  /\  Fun  { <. x ,  ( G `  Z )
>. } )  /\  ( dom  P  i^i  dom  { <. x ,  ( G `
 Z ) >. } )  =  (/) )  ->  Fun  ( P  u.  { <. x ,  ( G `  Z )
>. } ) )
295, 27, 28syl2anc 661 . 2  |-  ( ch 
->  Fun  ( P  u.  {
<. x ,  ( G `
 Z ) >. } ) )
30 bnj1421.12 . . 3  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
3130funeqi 5614 . 2  |-  ( Fun 
Q  <->  Fun  ( P  u.  {
<. x ,  ( G `
 Z ) >. } ) )
3229, 31sylibr 212 1  |-  ( ch 
->  Fun  Q )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819   {cab 2442    =/= wne 2652   A.wral 2807   E.wrex 2808   {crab 2811   [.wsbc 3327    u. cun 3469    i^i cin 3470    C_ wss 3471   (/)c0 3793   {csn 4032   <.cop 4038   U.cuni 4251   U_ciun 4332   class class class wbr 4456   dom cdm 5008    |` cres 5010   Fun wfun 5588    Fn wfn 5589   ` cfv 5594    predc-bnj14 33883    FrSe w-bnj15 33887    trClc-bnj18 33889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-reg 8036  ax-inf2 8075
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-1o 7148  df-bnj17 33882  df-bnj14 33884  df-bnj13 33886  df-bnj15 33888  df-bnj18 33890  df-bnj19 33892
This theorem is referenced by:  bnj1312  34257
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