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Theorem bnj1421 29411
Description: Technical lemma for bnj60 29431. 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 2959 . . . . 5  |-  x  e. 
_V
3 fvex 5742 . . . . 5  |-  ( G `
 Z )  e. 
_V
42, 3funsn 5499 . . . 4  |-  Fun  { <. x ,  ( G `
 Z ) >. }
51, 4jctir 525 . . 3  |-  ( ch 
->  ( Fun  P  /\  Fun  { <. x ,  ( G `  Z )
>. } ) )
6 bnj1421.15 . . . . 5  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
73dmsnop 5344 . . . . . 6  |-  dom  { <. x ,  ( G `
 Z ) >. }  =  { x }
87a1i 11 . . . . 5  |-  ( ch 
->  dom  { <. x ,  ( G `  Z ) >. }  =  { x } )
96, 8ineq12d 3543 . . . 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 447 . . . . . . 7  |-  ( ps 
->  R  FrSe  A )
1310, 12bnj835 29128 . . . . . 6  |-  ( ch 
->  R  FrSe  A )
14 biid 228 . . . . . . . 8  |-  ( R 
FrSe  A  <->  R  FrSe  A )
15 biid 228 . . . . . . . 8  |-  ( -.  x  e.  trCl (
x ,  A ,  R )  <->  -.  x  e.  trCl ( x ,  A ,  R ) )
16 biid 228 . . . . . . . 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 228 . . . . . . . 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 2436 . . . . . . . 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 29410 . . . . . . 7  |-  ( R 
FrSe  A  ->  A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R ) )
20 disjsn 3868 . . . . . . . 8  |-  ( ( 
trCl ( x ,  A ,  R )  i^i  { x }
)  =  (/)  <->  -.  x  e.  trCl ( x ,  A ,  R ) )
2120ralbii 2729 . . . . . . 7  |-  ( A. x  e.  A  (  trCl ( x ,  A ,  R )  i^i  {
x } )  =  (/) 
<-> 
A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R ) )
2219, 21sylibr 204 . . . . . 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 29171 . . . . 5  |-  ( ch 
->  x  e.  A
)
2623, 25bnj1294 29189 . . . 4  |-  ( ch 
->  (  trCl ( x ,  A ,  R
)  i^i  { x } )  =  (/) )
279, 26eqtrd 2468 . . 3  |-  ( ch 
->  ( dom  P  i^i  dom 
{ <. x ,  ( G `  Z )
>. } )  =  (/) )
28 funun 5495 . . 3  |-  ( ( ( Fun  P  /\  Fun  { <. x ,  ( G `  Z )
>. } )  /\  ( dom  P  i^i  dom  { <. x ,  ( G `
 Z ) >. } )  =  (/) )  ->  Fun  ( P  u.  { <. x ,  ( G `  Z )
>. } ) )
295, 27, 28syl2anc 643 . 2  |-  ( ch 
->  Fun  ( P  u.  {
<. x ,  ( G `
 Z ) >. } ) )
30 bnj1421.12 . . 3  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
3130funeqi 5474 . 2  |-  ( Fun 
Q  <->  Fun  ( P  u.  {
<. x ,  ( G `
 Z ) >. } ) )
3229, 31sylibr 204 1  |-  ( ch 
->  Fun  Q )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2422    =/= wne 2599   A.wral 2705   E.wrex 2706   {crab 2709   [.wsbc 3161    u. cun 3318    i^i cin 3319    C_ wss 3320   (/)c0 3628   {csn 3814   <.cop 3817   U.cuni 4015   U_ciun 4093   class class class wbr 4212   dom cdm 4878    |` cres 4880   Fun wfun 5448    Fn wfn 5449   ` cfv 5454    predc-bnj14 29052    FrSe w-bnj15 29056    trClc-bnj18 29058
This theorem is referenced by:  bnj1312  29427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-reg 7560  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-bnj17 29051  df-bnj14 29053  df-bnj13 29055  df-bnj15 29057  df-bnj18 29059  df-bnj19 29061
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