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Theorem bnj1321 29829
Description: Technical lemma for bnj60 29864. 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
bnj1321.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1321.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1321.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1321.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
Assertion
Ref Expression
bnj1321  |-  ( ( R  FrSe  A  /\  E. f ta )  ->  E! f ta )
Distinct variable groups:    A, d,
f, x    B, f    G, d, f    R, d, f, x
Allowed substitution hints:    ta( x, f, d)    B( x, d)    C( x, f, d)    G( x)    Y( x, f, d)

Proof of Theorem bnj1321
Dummy variables  g 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 463 . 2  |-  ( ( R  FrSe  A  /\  E. f ta )  ->  E. f ta )
2 simp1 1007 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  R  FrSe  A )
3 bnj1321.4 . . . . . . . . 9  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
43simplbi 462 . . . . . . . 8  |-  ( ta 
->  f  e.  C
)
543ad2ant2 1029 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  f  e.  C )
6 bnj1321.3 . . . . . . . . . . . . 13  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
7 nfab1 2593 . . . . . . . . . . . . 13  |-  F/_ f { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
86, 7nfcxfr 2589 . . . . . . . . . . . 12  |-  F/_ f C
98nfcri 2585 . . . . . . . . . . 11  |-  F/ f  g  e.  C
10 nfv 1760 . . . . . . . . . . 11  |-  F/ f dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
119, 10nfan 2010 . . . . . . . . . 10  |-  F/ f ( g  e.  C  /\  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
12 eleq1 2516 . . . . . . . . . . . 12  |-  ( f  =  g  ->  (
f  e.  C  <->  g  e.  C ) )
13 dmeq 5034 . . . . . . . . . . . . 13  |-  ( f  =  g  ->  dom  f  =  dom  g )
1413eqeq1d 2452 . . . . . . . . . . . 12  |-  ( f  =  g  ->  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  <->  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
1512, 14anbi12d 716 . . . . . . . . . . 11  |-  ( f  =  g  ->  (
( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )  <-> 
( g  e.  C  /\  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) ) )
163, 15syl5bb 261 . . . . . . . . . 10  |-  ( f  =  g  ->  ( ta 
<->  ( g  e.  C  /\  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) ) )
1711, 16sbie 2236 . . . . . . . . 9  |-  ( [ g  /  f ] ta  <->  ( g  e.  C  /\  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R
) ) ) )
1817simplbi 462 . . . . . . . 8  |-  ( [ g  /  f ] ta  ->  g  e.  C )
19183ad2ant3 1030 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  g  e.  C )
20 bnj1321.1 . . . . . . . 8  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
21 bnj1321.2 . . . . . . . 8  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
22 eqid 2450 . . . . . . . 8  |-  ( dom  f  i^i  dom  g
)  =  ( dom  f  i^i  dom  g
)
2320, 21, 6, 22bnj1326 29828 . . . . . . 7  |-  ( ( R  FrSe  A  /\  f  e.  C  /\  g  e.  C )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  ( dom  f  i^i  dom  g )
) )
242, 5, 19, 23syl3anc 1267 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  ( dom  f  i^i  dom  g )
) )
253simprbi 466 . . . . . . . . . 10  |-  ( ta 
->  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
26253ad2ant2 1029 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
2717simprbi 466 . . . . . . . . . 10  |-  ( [ g  /  f ] ta  ->  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
28273ad2ant3 1030 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
2926, 28eqtr4d 2487 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  dom  f  =  dom  g )
30 bnj1322 29627 . . . . . . . . 9  |-  ( dom  f  =  dom  g  ->  ( dom  f  i^i 
dom  g )  =  dom  f )
3130reseq2d 5104 . . . . . . . 8  |-  ( dom  f  =  dom  g  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( f  |`  dom  f ) )
3229, 31syl 17 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( f  |`  dom  f ) )
33 releq 4916 . . . . . . . . 9  |-  ( z  =  f  ->  ( Rel  z  <->  Rel  f ) )
3420, 21, 6bnj66 29664 . . . . . . . . 9  |-  ( z  e.  C  ->  Rel  z )
3533, 34vtoclga 3112 . . . . . . . 8  |-  ( f  e.  C  ->  Rel  f )
36 resdm 5145 . . . . . . . 8  |-  ( Rel  f  ->  ( f  |` 
dom  f )  =  f )
375, 35, 363syl 18 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( f  |`  dom  f
)  =  f )
3832, 37eqtrd 2484 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  f )
39 eqeq2 2461 . . . . . . . . . 10  |-  ( dom  f  =  dom  g  ->  ( ( dom  f  i^i  dom  g )  =  dom  f  <->  ( dom  f  i^i  dom  g )  =  dom  g ) )
4030, 39mpbid 214 . . . . . . . . 9  |-  ( dom  f  =  dom  g  ->  ( dom  f  i^i 
dom  g )  =  dom  g )
4140reseq2d 5104 . . . . . . . 8  |-  ( dom  f  =  dom  g  ->  ( g  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  dom  g ) )
4229, 41syl 17 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( g  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  dom  g ) )
4320, 21, 6bnj66 29664 . . . . . . . 8  |-  ( g  e.  C  ->  Rel  g )
44 resdm 5145 . . . . . . . 8  |-  ( Rel  g  ->  ( g  |` 
dom  g )  =  g )
4519, 43, 443syl 18 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( g  |`  dom  g
)  =  g )
4642, 45eqtrd 2484 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( g  |`  ( dom  f  i^i  dom  g
) )  =  g )
4724, 38, 463eqtr3d 2492 . . . . 5  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  f  =  g )
48473expib 1210 . . . 4  |-  ( R 
FrSe  A  ->  ( ( ta  /\  [ g  /  f ] ta )  ->  f  =  g ) )
4948alrimivv 1773 . . 3  |-  ( R 
FrSe  A  ->  A. f A. g ( ( ta 
/\  [ g  / 
f ] ta )  ->  f  =  g ) )
5049adantr 467 . 2  |-  ( ( R  FrSe  A  /\  E. f ta )  ->  A. f A. g ( ( ta  /\  [
g  /  f ] ta )  ->  f  =  g ) )
51 nfv 1760 . . 3  |-  F/ g ta
5251eu2 2337 . 2  |-  ( E! f ta  <->  ( E. f ta  /\  A. f A. g ( ( ta 
/\  [ g  / 
f ] ta )  ->  f  =  g ) ) )
531, 50, 52sylanbrc 669 1  |-  ( ( R  FrSe  A  /\  E. f ta )  ->  E! f ta )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984   A.wal 1441    = wceq 1443   E.wex 1662   [wsb 1796    e. wcel 1886   E!weu 2298   {cab 2436   A.wral 2736   E.wrex 2737    u. cun 3401    i^i cin 3402    C_ wss 3403   {csn 3967   <.cop 3973   dom cdm 4833    |` cres 4835   Rel wrel 4838    Fn wfn 5576   ` cfv 5581    predc-bnj14 29486    FrSe w-bnj15 29490    trClc-bnj18 29492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-reg 8104  ax-inf2 8143
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6690  df-1o 7179  df-bnj17 29485  df-bnj14 29487  df-bnj13 29489  df-bnj15 29491  df-bnj18 29493  df-bnj19 29495
This theorem is referenced by:  bnj1489  29858
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