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Theorem bnj1321 32015
Description: Technical lemma for bnj60 32050. 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 461 . 2  |-  ( ( R  FrSe  A  /\  E. f ta )  ->  E. f ta )
2 simp1 988 . . . . . . 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 460 . . . . . . . 8  |-  ( ta 
->  f  e.  C
)
543ad2ant2 1010 . . . . . . 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 2579 . . . . . . . . . . . . 13  |-  F/_ f { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
86, 7nfcxfr 2574 . . . . . . . . . . . 12  |-  F/_ f C
98nfcri 2571 . . . . . . . . . . 11  |-  F/ f  g  e.  C
10 nfv 1673 . . . . . . . . . . 11  |-  F/ f dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
119, 10nfan 1861 . . . . . . . . . 10  |-  F/ f ( g  e.  C  /\  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
12 eleq1 2501 . . . . . . . . . . . 12  |-  ( f  =  g  ->  (
f  e.  C  <->  g  e.  C ) )
13 dmeq 5038 . . . . . . . . . . . . 13  |-  ( f  =  g  ->  dom  f  =  dom  g )
1413eqeq1d 2449 . . . . . . . . . . . 12  |-  ( f  =  g  ->  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  <->  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
1512, 14anbi12d 710 . . . . . . . . . . 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 257 . . . . . . . . . 10  |-  ( f  =  g  ->  ( ta 
<->  ( g  e.  C  /\  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) ) )
1711, 16sbie 2100 . . . . . . . . 9  |-  ( [ g  /  f ] ta  <->  ( g  e.  C  /\  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R
) ) ) )
1817simplbi 460 . . . . . . . 8  |-  ( [ g  /  f ] ta  ->  g  e.  C )
19183ad2ant3 1011 . . . . . . 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 2441 . . . . . . . 8  |-  ( dom  f  i^i  dom  g
)  =  ( dom  f  i^i  dom  g
)
2320, 21, 6, 22bnj1326 32014 . . . . . . 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 1218 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  ( dom  f  i^i  dom  g )
) )
253simprbi 464 . . . . . . . . . 10  |-  ( ta 
->  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
26253ad2ant2 1010 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
2717simprbi 464 . . . . . . . . . 10  |-  ( [ g  /  f ] ta  ->  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
28273ad2ant3 1011 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
2926, 28eqtr4d 2476 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  dom  f  =  dom  g )
30 bnj1322 31813 . . . . . . . . 9  |-  ( dom  f  =  dom  g  ->  ( dom  f  i^i 
dom  g )  =  dom  f )
3130reseq2d 5108 . . . . . . . 8  |-  ( dom  f  =  dom  g  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( f  |`  dom  f ) )
3229, 31syl 16 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( f  |`  dom  f ) )
33 releq 4920 . . . . . . . . 9  |-  ( z  =  f  ->  ( Rel  z  <->  Rel  f ) )
3420, 21, 6bnj66 31850 . . . . . . . . 9  |-  ( z  e.  C  ->  Rel  z )
3533, 34vtoclga 3034 . . . . . . . 8  |-  ( f  e.  C  ->  Rel  f )
36 resdm 5146 . . . . . . . 8  |-  ( Rel  f  ->  ( f  |` 
dom  f )  =  f )
375, 35, 363syl 20 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( f  |`  dom  f
)  =  f )
3832, 37eqtrd 2473 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  f )
39 eqeq2 2450 . . . . . . . . . 10  |-  ( dom  f  =  dom  g  ->  ( ( dom  f  i^i  dom  g )  =  dom  f  <->  ( dom  f  i^i  dom  g )  =  dom  g ) )
4030, 39mpbid 210 . . . . . . . . 9  |-  ( dom  f  =  dom  g  ->  ( dom  f  i^i 
dom  g )  =  dom  g )
4140reseq2d 5108 . . . . . . . 8  |-  ( dom  f  =  dom  g  ->  ( g  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  dom  g ) )
4229, 41syl 16 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( g  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  dom  g ) )
4320, 21, 6bnj66 31850 . . . . . . . 8  |-  ( g  e.  C  ->  Rel  g )
44 resdm 5146 . . . . . . . 8  |-  ( Rel  g  ->  ( g  |` 
dom  g )  =  g )
4519, 43, 443syl 20 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( g  |`  dom  g
)  =  g )
4642, 45eqtrd 2473 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( g  |`  ( dom  f  i^i  dom  g
) )  =  g )
4724, 38, 463eqtr3d 2481 . . . . 5  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  f  =  g )
48473expib 1190 . . . 4  |-  ( R 
FrSe  A  ->  ( ( ta  /\  [ g  /  f ] ta )  ->  f  =  g ) )
4948alrimivv 1686 . . 3  |-  ( R 
FrSe  A  ->  A. f A. g ( ( ta 
/\  [ g  / 
f ] ta )  ->  f  =  g ) )
5049adantr 465 . 2  |-  ( ( R  FrSe  A  /\  E. f ta )  ->  A. f A. g ( ( ta  /\  [
g  /  f ] ta )  ->  f  =  g ) )
51 nfv 1673 . . 3  |-  F/ g ta
5251eu2 2302 . 2  |-  ( E! f ta  <->  ( E. f ta  /\  A. f A. g ( ( ta 
/\  [ g  / 
f ] ta )  ->  f  =  g ) ) )
531, 50, 52sylanbrc 664 1  |-  ( ( R  FrSe  A  /\  E. f ta )  ->  E! f ta )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369   E.wex 1586   [wsb 1700    e. wcel 1756   E!weu 2253   {cab 2427   A.wral 2713   E.wrex 2714    u. cun 3324    i^i cin 3325    C_ wss 3326   {csn 3875   <.cop 3881   dom cdm 4838    |` cres 4840   Rel wrel 4843    Fn wfn 5411   ` cfv 5416    predc-bnj14 31673    FrSe w-bnj15 31677    trClc-bnj18 31679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-reg 7805  ax-inf2 7845
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-om 6475  df-1o 6918  df-bnj17 31672  df-bnj14 31674  df-bnj13 31676  df-bnj15 31678  df-bnj18 31680  df-bnj19 31682
This theorem is referenced by:  bnj1489  32044
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