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Theorem bnj1493 29502
Description: Technical lemma for bnj60 29505. 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
bnj1493.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1493.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1493.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
Assertion
Ref Expression
bnj1493  |-  ( R 
FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
Distinct variable groups:    A, d,
f, x    B, f    G, d, f, x    R, d, f, x
Allowed substitution hints:    B( x, d)    C( x, f, d)    Y( x, f, d)

Proof of Theorem bnj1493
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj1493.1 . 2  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
2 bnj1493.2 . 2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
3 bnj1493.3 . 2  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
4 biid 229 . 2  |-  ( ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )  <-> 
( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
5 eqid 2438 . 2  |-  { x  e.  A  |  -.  E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R
) ) ) }  =  { x  e.  A  |  -.  E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R
) ) ) }
6 biid 229 . 2  |-  ( ( R  FrSe  A  /\  { x  e.  A  |  -.  E. f ( f  e.  C  /\  dom  f  =  ( {
x }  u.  trCl ( x ,  A ,  R ) ) ) }  =/=  (/) )  <->  ( R  FrSe  A  /\  { x  e.  A  |  -.  E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R
) ) ) }  =/=  (/) ) )
7 biid 229 . 2  |-  ( ( ( R  FrSe  A  /\  { x  e.  A  |  -.  E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) }  =/=  (/) )  /\  x  e.  { x  e.  A  |  -.  E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R
) ) ) }  /\  A. y  e. 
{ x  e.  A  |  -.  E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) }  -.  y R x )  <->  ( ( R  FrSe  A  /\  {
x  e.  A  |  -.  E. f ( f  e.  C  /\  dom  f  =  ( {
x }  u.  trCl ( x ,  A ,  R ) ) ) }  =/=  (/) )  /\  x  e.  { x  e.  A  |  -.  E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R
) ) ) }  /\  A. y  e. 
{ x  e.  A  |  -.  E. f ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) }  -.  y R x ) )
8 biid 229 . 2  |-  ( [. y  /  x ]. (
f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )  <->  [. y  /  x ]. ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
9 eqid 2438 . 2  |-  { f  |  E. y  e. 
pred  ( x ,  A ,  R )
[. y  /  x ]. ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) }  =  { f  |  E. y  e. 
pred  ( x ,  A ,  R )
[. y  /  x ]. ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) }
10 eqid 2438 . 2  |-  U. {
f  |  E. y  e.  pred  ( x ,  A ,  R )
[. y  /  x ]. ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) }  =  U. {
f  |  E. y  e.  pred  ( x ,  A ,  R )
[. y  /  x ]. ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) }
11 eqid 2438 . 2  |-  <. x ,  ( U. {
f  |  E. y  e.  pred  ( x ,  A ,  R )
[. y  /  x ]. ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) }  |`  pred ( x ,  A ,  R
) ) >.  =  <. x ,  ( U. {
f  |  E. y  e.  pred  ( x ,  A ,  R )
[. y  /  x ]. ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) }  |`  pred ( x ,  A ,  R
) ) >.
12 eqid 2438 . 2  |-  ( U. { f  |  E. y  e.  pred  ( x ,  A ,  R
) [. y  /  x ]. ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) }  u.  { <. x ,  ( G `  <. x ,  ( U. { f  |  E. y  e.  pred  ( x ,  A ,  R
) [. y  /  x ]. ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) }  |`  pred ( x ,  A ,  R
) ) >. ) >. } )  =  ( U. { f  |  E. y  e.  pred  ( x ,  A ,  R ) [. y  /  x ]. ( f  e.  C  /\  dom  f  =  ( {
x }  u.  trCl ( x ,  A ,  R ) ) ) }  u.  { <. x ,  ( G `  <. x ,  ( U. { f  |  E. y  e.  pred  ( x ,  A ,  R
) [. y  /  x ]. ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) }  |`  pred ( x ,  A ,  R
) ) >. ) >. } )
13 eqid 2438 . 2  |-  <. z ,  ( ( U. { f  |  E. y  e.  pred  ( x ,  A ,  R
) [. y  /  x ]. ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) }  u.  { <. x ,  ( G `  <. x ,  ( U. { f  |  E. y  e.  pred  ( x ,  A ,  R
) [. y  /  x ]. ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) }  |`  pred ( x ,  A ,  R
) ) >. ) >. } )  |`  pred (
z ,  A ,  R ) ) >.  =  <. z ,  ( ( U. { f  |  E. y  e. 
pred  ( x ,  A ,  R )
[. y  /  x ]. ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) }  u.  { <. x ,  ( G `  <. x ,  ( U. { f  |  E. y  e.  pred  ( x ,  A ,  R
) [. y  /  x ]. ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) }  |`  pred ( x ,  A ,  R
) ) >. ) >. } )  |`  pred (
z ,  A ,  R ) ) >.
14 eqid 2438 . 2  |-  ( { x }  u.  trCl ( x ,  A ,  R ) )  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14bnj1312 29501 1  |-  ( R 
FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ 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    C_ wss 3322   (/)c0 3630   {csn 3816   <.cop 3819   U.cuni 4017   class class class wbr 4215   dom cdm 4881    |` cres 4883    Fn wfn 5452   ` cfv 5457    predc-bnj14 29126    FrSe w-bnj15 29130    trClc-bnj18 29132
This theorem is referenced by:  bnj1498  29504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-reg 7563  ax-inf2 7599
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 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-1o 6727  df-bnj17 29125  df-bnj14 29127  df-bnj13 29129  df-bnj15 29131  df-bnj18 29133  df-bnj19 29135
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