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Theorem bnj1321 29398
Description: Technical lemma for bnj60 29433. 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 449 . 2  |-  ( ( R  FrSe  A  /\  E. f ta )  ->  E. f ta )
2 simp1 958 . . . . . . 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 448 . . . . . . . 8  |-  ( ta 
->  f  e.  C
)
543ad2ant2 980 . . . . . . 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 2576 . . . . . . . . . . . . 13  |-  F/_ f { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
86, 7nfcxfr 2571 . . . . . . . . . . . 12  |-  F/_ f C
98nfcri 2568 . . . . . . . . . . 11  |-  F/ f  g  e.  C
10 nfv 1630 . . . . . . . . . . 11  |-  F/ f dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
119, 10nfan 1847 . . . . . . . . . 10  |-  F/ f ( g  e.  C  /\  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
12 eleq1 2498 . . . . . . . . . . . 12  |-  ( f  =  g  ->  (
f  e.  C  <->  g  e.  C ) )
13 dmeq 5072 . . . . . . . . . . . . 13  |-  ( f  =  g  ->  dom  f  =  dom  g )
1413eqeq1d 2446 . . . . . . . . . . . 12  |-  ( f  =  g  ->  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  <->  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
1512, 14anbi12d 693 . . . . . . . . . . 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 250 . . . . . . . . . 10  |-  ( f  =  g  ->  ( ta 
<->  ( g  e.  C  /\  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) ) )
1711, 16sbie 2151 . . . . . . . . 9  |-  ( [ g  /  f ] ta  <->  ( g  e.  C  /\  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R
) ) ) )
1817simplbi 448 . . . . . . . 8  |-  ( [ g  /  f ] ta  ->  g  e.  C )
19183ad2ant3 981 . . . . . . 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 2438 . . . . . . . 8  |-  ( dom  f  i^i  dom  g
)  =  ( dom  f  i^i  dom  g
)
2320, 21, 6, 22bnj1326 29397 . . . . . . 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 1185 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  ( dom  f  i^i  dom  g )
) )
253simprbi 452 . . . . . . . . . 10  |-  ( ta 
->  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
26253ad2ant2 980 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
2717simprbi 452 . . . . . . . . . 10  |-  ( [ g  /  f ] ta  ->  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
28273ad2ant3 981 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
2926, 28eqtr4d 2473 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  dom  f  =  dom  g )
30 bnj1322 29196 . . . . . . . . 9  |-  ( dom  f  =  dom  g  ->  ( dom  f  i^i 
dom  g )  =  dom  f )
3130reseq2d 5148 . . . . . . . 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 4961 . . . . . . . . 9  |-  ( z  =  f  ->  ( Rel  z  <->  Rel  f ) )
3420, 21, 6bnj66 29233 . . . . . . . . 9  |-  ( z  e.  C  ->  Rel  z )
3533, 34vtoclga 3019 . . . . . . . 8  |-  ( f  e.  C  ->  Rel  f )
36 resdm 5186 . . . . . . . 8  |-  ( Rel  f  ->  ( f  |` 
dom  f )  =  f )
375, 35, 363syl 19 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( f  |`  dom  f
)  =  f )
3832, 37eqtrd 2470 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  f )
39 eqeq2 2447 . . . . . . . . . 10  |-  ( dom  f  =  dom  g  ->  ( ( dom  f  i^i  dom  g )  =  dom  f  <->  ( dom  f  i^i  dom  g )  =  dom  g ) )
4030, 39mpbid 203 . . . . . . . . 9  |-  ( dom  f  =  dom  g  ->  ( dom  f  i^i 
dom  g )  =  dom  g )
4140reseq2d 5148 . . . . . . . 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 29233 . . . . . . . 8  |-  ( g  e.  C  ->  Rel  g )
44 resdm 5186 . . . . . . . 8  |-  ( Rel  g  ->  ( g  |` 
dom  g )  =  g )
4519, 43, 443syl 19 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( g  |`  dom  g
)  =  g )
4642, 45eqtrd 2470 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( g  |`  ( dom  f  i^i  dom  g
) )  =  g )
4724, 38, 463eqtr3d 2478 . . . . 5  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  f  =  g )
48473expib 1157 . . . 4  |-  ( R 
FrSe  A  ->  ( ( ta  /\  [ g  /  f ] ta )  ->  f  =  g ) )
4948alrimivv 1643 . . 3  |-  ( R 
FrSe  A  ->  A. f A. g ( ( ta 
/\  [ g  / 
f ] ta )  ->  f  =  g ) )
5049adantr 453 . 2  |-  ( ( R  FrSe  A  /\  E. f ta )  ->  A. f A. g ( ( ta  /\  [
g  /  f ] ta )  ->  f  =  g ) )
51 nfv 1630 . . 3  |-  F/ g ta
5251eu2 2308 . 2  |-  ( E! f ta  <->  ( E. f ta  /\  A. f A. g ( ( ta 
/\  [ g  / 
f ] ta )  ->  f  =  g ) ) )
531, 50, 52sylanbrc 647 1  |-  ( ( R  FrSe  A  /\  E. f ta )  ->  E! f ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   A.wal 1550   E.wex 1551    = wceq 1653   [wsb 1659    e. wcel 1726   E!weu 2283   {cab 2424   A.wral 2707   E.wrex 2708    u. cun 3320    i^i cin 3321    C_ wss 3322   {csn 3816   <.cop 3819   dom cdm 4880    |` cres 4882   Rel wrel 4885    Fn wfn 5451   ` cfv 5456    predc-bnj14 29054    FrSe w-bnj15 29058    trClc-bnj18 29060
This theorem is referenced by:  bnj1489  29427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-reg 7562  ax-inf2 7598
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 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-1o 6726  df-bnj17 29053  df-bnj14 29055  df-bnj13 29057  df-bnj15 29059  df-bnj18 29061  df-bnj19 29063
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