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Theorem bnj1421 29388
Description: Technical lemma for bnj60 29408. 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 2804 . . . . 5  |-  x  e. 
_V
3 fvex 5555 . . . . 5  |-  ( G `
 Z )  e. 
_V
42, 3funsn 5316 . . . 4  |-  Fun  { <. x ,  ( G `
 Z ) >. }
51, 4jctir 524 . . 3  |-  ( ch 
->  ( Fun  P  /\  Fun  { <. x ,  ( G `  Z )
>. } ) )
6 bnj1421.15 . . . . 5  |-  ( ch 
->  dom  P  =  trCl ( x ,  A ,  R ) )
73dmsnop 5163 . . . . . 6  |-  dom  { <. x ,  ( G `
 Z ) >. }  =  { x }
87a1i 10 . . . . 5  |-  ( ch 
->  dom  { <. x ,  ( G `  Z ) >. }  =  { x } )
96, 8ineq12d 3384 . . . 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 446 . . . . . . 7  |-  ( ps 
->  R  FrSe  A )
1310, 12bnj835 29105 . . . . . 6  |-  ( ch 
->  R  FrSe  A )
14 biid 227 . . . . . . . 8  |-  ( R 
FrSe  A  <->  R  FrSe  A )
15 biid 227 . . . . . . . 8  |-  ( -.  x  e.  trCl (
x ,  A ,  R )  <->  -.  x  e.  trCl ( x ,  A ,  R ) )
16 biid 227 . . . . . . . 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 227 . . . . . . . 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 2296 . . . . . . . 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 29387 . . . . . . 7  |-  ( R 
FrSe  A  ->  A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R ) )
20 disjsn 3706 . . . . . . . 8  |-  ( ( 
trCl ( x ,  A ,  R )  i^i  { x }
)  =  (/)  <->  -.  x  e.  trCl ( x ,  A ,  R ) )
2120ralbii 2580 . . . . . . 7  |-  ( A. x  e.  A  (  trCl ( x ,  A ,  R )  i^i  {
x } )  =  (/) 
<-> 
A. x  e.  A  -.  x  e.  trCl ( x ,  A ,  R ) )
2219, 21sylibr 203 . . . . . 6  |-  ( R 
FrSe  A  ->  A. x  e.  A  (  trCl ( x ,  A ,  R )  i^i  {
x } )  =  (/) )
2313, 22syl 15 . . . . 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 29148 . . . . 5  |-  ( ch 
->  x  e.  A
)
2623, 25bnj1294 29166 . . . 4  |-  ( ch 
->  (  trCl ( x ,  A ,  R
)  i^i  { x } )  =  (/) )
279, 26eqtrd 2328 . . 3  |-  ( ch 
->  ( dom  P  i^i  dom 
{ <. x ,  ( G `  Z )
>. } )  =  (/) )
28 funun 5312 . . 3  |-  ( ( ( Fun  P  /\  Fun  { <. x ,  ( G `  Z )
>. } )  /\  ( dom  P  i^i  dom  { <. x ,  ( G `
 Z ) >. } )  =  (/) )  ->  Fun  ( P  u.  { <. x ,  ( G `  Z )
>. } ) )
295, 27, 28syl2anc 642 . 2  |-  ( ch 
->  Fun  ( P  u.  {
<. x ,  ( G `
 Z ) >. } ) )
30 bnj1421.12 . . 3  |-  Q  =  ( P  u.  { <. x ,  ( G `
 Z ) >. } )
3130funeqi 5291 . 2  |-  ( Fun 
Q  <->  Fun  ( P  u.  {
<. x ,  ( G `
 Z ) >. } ) )
3229, 31sylibr 203 1  |-  ( ch 
->  Fun  Q )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560   [.wsbc 3004    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   <.cop 3656   U.cuni 3843   U_ciun 3921   class class class wbr 4039   dom cdm 4705    |` cres 4707   Fun wfun 5265    Fn wfn 5266   ` cfv 5271    predc-bnj14 29029    FrSe w-bnj15 29033    trClc-bnj18 29035
This theorem is referenced by:  bnj1312  29404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-bnj17 29028  df-bnj14 29030  df-bnj13 29032  df-bnj15 29034  df-bnj18 29036  df-bnj19 29038
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