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Theorem bnj1493 28834
Description: Technical lemma for bnj60 28837. 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 227 . 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 2358 . 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 227 . 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 227 . 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 227 . 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 2358 . 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 2358 . 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 2358 . 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 2358 . 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 2358 . 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 2358 . 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 28833 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 358    /\ w3a 934   E.wex 1541    = wceq 1642    e. wcel 1710   {cab 2344    =/= wne 2521   A.wral 2619   E.wrex 2620   {crab 2623   [.wsbc 3067    u. cun 3226    C_ wss 3228   (/)c0 3531   {csn 3716   <.cop 3719   U.cuni 3906   class class class wbr 4102   dom cdm 4768    |` cres 4770    Fn wfn 5329   ` cfv 5334    predc-bnj14 28458    FrSe w-bnj15 28462    trClc-bnj18 28464
This theorem is referenced by:  bnj1498  28836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-reg 7393  ax-inf2 7429
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-1o 6563  df-bnj17 28457  df-bnj14 28459  df-bnj13 28461  df-bnj15 28463  df-bnj18 28465  df-bnj19 28467
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