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Theorem bnj1421 29473
Description: Technical lemma for bnj60 29493. 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 2961 . . . . 5  |-  x  e. 
_V
3 fvex 5744 . . . . 5  |-  ( G `
 Z )  e. 
_V
42, 3funsn 5501 . . . 4  |-  Fun  { <. x ,  ( G `
 Z ) >. }
51, 4jctir 526 . . 3  |-  ( ch 
->  ( Fun  P  /\  Fun  { <. x ,  ( G `  Z )
>. } ) )
6 bnj1421.15 . . . . 5  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
73dmsnop 5346 . . . . . 6  |-  dom  { <. x ,  ( G `
 Z ) >. }  =  { x }
87a1i 11 . . . . 5  |-  ( ch 
->  dom  { <. x ,  ( G `  Z ) >. }  =  { x } )
96, 8ineq12d 3545 . . . 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 448 . . . . . . 7  |-  ( ps 
->  R  FrSe  A )
1310, 12bnj835 29190 . . . . . 6  |-  ( ch 
->  R  FrSe  A )
14 biid 229 . . . . . . . 8  |-  ( R 
FrSe  A  <->  R  FrSe  A )
15 biid 229 . . . . . . . 8  |-  ( -.  x  e.  trCl (
x ,  A ,  R )  <->  -.  x  e.  trCl ( x ,  A ,  R ) )
16 biid 229 . . . . . . . 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 229 . . . . . . . 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 2438 . . . . . . . 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 29472 . . . . . . 7  |-  ( R 
FrSe  A  ->  A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R ) )
20 disjsn 3870 . . . . . . . 8  |-  ( ( 
trCl ( x ,  A ,  R )  i^i  { x }
)  =  (/)  <->  -.  x  e.  trCl ( x ,  A ,  R ) )
2120ralbii 2731 . . . . . . 7  |-  ( A. x  e.  A  (  trCl ( x ,  A ,  R )  i^i  {
x } )  =  (/) 
<-> 
A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R ) )
2219, 21sylibr 205 . . . . . 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 29233 . . . . 5  |-  ( ch 
->  x  e.  A
)
2623, 25bnj1294 29251 . . . 4  |-  ( ch 
->  (  trCl ( x ,  A ,  R
)  i^i  { x } )  =  (/) )
279, 26eqtrd 2470 . . 3  |-  ( ch 
->  ( dom  P  i^i  dom 
{ <. x ,  ( G `  Z )
>. } )  =  (/) )
28 funun 5497 . . 3  |-  ( ( ( Fun  P  /\  Fun  { <. x ,  ( G `  Z )
>. } )  /\  ( dom  P  i^i  dom  { <. x ,  ( G `
 Z ) >. } )  =  (/) )  ->  Fun  ( P  u.  { <. x ,  ( G `  Z )
>. } ) )
295, 27, 28syl2anc 644 . 2  |-  ( ch 
->  Fun  ( P  u.  {
<. x ,  ( G `
 Z ) >. } ) )
30 bnj1421.12 . . 3  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
3130funeqi 5476 . 2  |-  ( Fun 
Q  <->  Fun  ( P  u.  {
<. x ,  ( G `
 Z ) >. } ) )
3229, 31sylibr 205 1  |-  ( ch 
->  Fun  Q )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711   [.wsbc 3163    u. cun 3320    i^i cin 3321    C_ wss 3322   (/)c0 3630   {csn 3816   <.cop 3819   U.cuni 4017   U_ciun 4095   class class class wbr 4214   dom cdm 4880    |` cres 4882   Fun wfun 5450    Fn wfn 5451   ` cfv 5456    predc-bnj14 29114    FrSe w-bnj15 29118    trClc-bnj18 29120
This theorem is referenced by:  bnj1312  29489
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-reg 7562  ax-inf2 7598
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-1o 6726  df-bnj17 29113  df-bnj14 29115  df-bnj13 29117  df-bnj15 29119  df-bnj18 29121  df-bnj19 29123
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