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Theorem bnj1321 29650
Description: Technical lemma for bnj60 29685. 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 462 . 2  |-  ( ( R  FrSe  A  /\  E. f ta )  ->  E. f ta )
2 simp1 1005 . . . . . . 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 461 . . . . . . . 8  |-  ( ta 
->  f  e.  C
)
543ad2ant2 1027 . . . . . . 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 2584 . . . . . . . . . . . . 13  |-  F/_ f { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
86, 7nfcxfr 2580 . . . . . . . . . . . 12  |-  F/_ f C
98nfcri 2575 . . . . . . . . . . 11  |-  F/ f  g  e.  C
10 nfv 1751 . . . . . . . . . . 11  |-  F/ f dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
119, 10nfan 1983 . . . . . . . . . 10  |-  F/ f ( g  e.  C  /\  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
12 eleq1 2492 . . . . . . . . . . . 12  |-  ( f  =  g  ->  (
f  e.  C  <->  g  e.  C ) )
13 dmeq 5046 . . . . . . . . . . . . 13  |-  ( f  =  g  ->  dom  f  =  dom  g )
1413eqeq1d 2422 . . . . . . . . . . . 12  |-  ( f  =  g  ->  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  <->  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
1512, 14anbi12d 715 . . . . . . . . . . 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 260 . . . . . . . . . 10  |-  ( f  =  g  ->  ( ta 
<->  ( g  e.  C  /\  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) ) )
1711, 16sbie 2200 . . . . . . . . 9  |-  ( [ g  /  f ] ta  <->  ( g  e.  C  /\  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R
) ) ) )
1817simplbi 461 . . . . . . . 8  |-  ( [ g  /  f ] ta  ->  g  e.  C )
19183ad2ant3 1028 . . . . . . 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 2420 . . . . . . . 8  |-  ( dom  f  i^i  dom  g
)  =  ( dom  f  i^i  dom  g
)
2320, 21, 6, 22bnj1326 29649 . . . . . . 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 1264 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  ( dom  f  i^i  dom  g )
) )
253simprbi 465 . . . . . . . . . 10  |-  ( ta 
->  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
26253ad2ant2 1027 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
2717simprbi 465 . . . . . . . . . 10  |-  ( [ g  /  f ] ta  ->  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
28273ad2ant3 1028 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
2926, 28eqtr4d 2464 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  dom  f  =  dom  g )
30 bnj1322 29448 . . . . . . . . 9  |-  ( dom  f  =  dom  g  ->  ( dom  f  i^i 
dom  g )  =  dom  f )
3130reseq2d 5116 . . . . . . . 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 4928 . . . . . . . . 9  |-  ( z  =  f  ->  ( Rel  z  <->  Rel  f ) )
3420, 21, 6bnj66 29485 . . . . . . . . 9  |-  ( z  e.  C  ->  Rel  z )
3533, 34vtoclga 3142 . . . . . . . 8  |-  ( f  e.  C  ->  Rel  f )
36 resdm 5157 . . . . . . . 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 2461 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  f )
39 eqeq2 2435 . . . . . . . . . 10  |-  ( dom  f  =  dom  g  ->  ( ( dom  f  i^i  dom  g )  =  dom  f  <->  ( dom  f  i^i  dom  g )  =  dom  g ) )
4030, 39mpbid 213 . . . . . . . . 9  |-  ( dom  f  =  dom  g  ->  ( dom  f  i^i 
dom  g )  =  dom  g )
4140reseq2d 5116 . . . . . . . 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 29485 . . . . . . . 8  |-  ( g  e.  C  ->  Rel  g )
44 resdm 5157 . . . . . . . 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 2461 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( g  |`  ( dom  f  i^i  dom  g
) )  =  g )
4724, 38, 463eqtr3d 2469 . . . . 5  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  f  =  g )
48473expib 1208 . . . 4  |-  ( R 
FrSe  A  ->  ( ( ta  /\  [ g  /  f ] ta )  ->  f  =  g ) )
4948alrimivv 1764 . . 3  |-  ( R 
FrSe  A  ->  A. f A. g ( ( ta 
/\  [ g  / 
f ] ta )  ->  f  =  g ) )
5049adantr 466 . 2  |-  ( ( R  FrSe  A  /\  E. f ta )  ->  A. f A. g ( ( ta  /\  [
g  /  f ] ta )  ->  f  =  g ) )
51 nfv 1751 . . 3  |-  F/ g ta
5251eu2 2303 . 2  |-  ( E! f ta  <->  ( E. f ta  /\  A. f A. g ( ( ta 
/\  [ g  / 
f ] ta )  ->  f  =  g ) ) )
531, 50, 52sylanbrc 668 1  |-  ( ( R  FrSe  A  /\  E. f ta )  ->  E! f ta )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437   E.wex 1659   [wsb 1786    e. wcel 1867   E!weu 2263   {cab 2405   A.wral 2773   E.wrex 2774    u. cun 3431    i^i cin 3432    C_ wss 3433   {csn 3993   <.cop 3999   dom cdm 4845    |` cres 4847   Rel wrel 4850    Fn wfn 5587   ` cfv 5592    predc-bnj14 29307    FrSe w-bnj15 29311    trClc-bnj18 29313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-reg 8098  ax-inf2 8137
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-om 6698  df-1o 7181  df-bnj17 29306  df-bnj14 29308  df-bnj13 29310  df-bnj15 29312  df-bnj18 29314  df-bnj19 29316
This theorem is referenced by:  bnj1489  29679
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