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Theorem bnj1321 29373
Description: Technical lemma for bnj60 29408. 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 447 . 2  |-  ( ( R  FrSe  A  /\  E. f ta )  ->  E. f ta )
2 simp1 955 . . . . . . 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 446 . . . . . . . 8  |-  ( ta 
->  f  e.  C
)
543ad2ant2 977 . . . . . . 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 2434 . . . . . . . . . . . . 13  |-  F/_ f { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
86, 7nfcxfr 2429 . . . . . . . . . . . 12  |-  F/_ f C
98nfcri 2426 . . . . . . . . . . 11  |-  F/ f  g  e.  C
10 nfv 1609 . . . . . . . . . . 11  |-  F/ f dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) )
119, 10nfan 1783 . . . . . . . . . 10  |-  F/ f ( g  e.  C  /\  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
12 eleq1 2356 . . . . . . . . . . . 12  |-  ( f  =  g  ->  (
f  e.  C  <->  g  e.  C ) )
13 dmeq 4895 . . . . . . . . . . . . 13  |-  ( f  =  g  ->  dom  f  =  dom  g )
1413eqeq1d 2304 . . . . . . . . . . . 12  |-  ( f  =  g  ->  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  <->  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
1512, 14anbi12d 691 . . . . . . . . . . 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 248 . . . . . . . . . 10  |-  ( f  =  g  ->  ( ta 
<->  ( g  e.  C  /\  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) ) )
1711, 16sbie 1991 . . . . . . . . 9  |-  ( [ g  /  f ] ta  <->  ( g  e.  C  /\  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R
) ) ) )
1817simplbi 446 . . . . . . . 8  |-  ( [ g  /  f ] ta  ->  g  e.  C )
19183ad2ant3 978 . . . . . . 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 2296 . . . . . . . 8  |-  ( dom  f  i^i  dom  g
)  =  ( dom  f  i^i  dom  g
)
2320, 21, 6, 22bnj1326 29372 . . . . . . 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 1182 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  ( dom  f  i^i  dom  g )
) )
253simprbi 450 . . . . . . . . . 10  |-  ( ta 
->  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
26253ad2ant2 977 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
2717simprbi 450 . . . . . . . . . 10  |-  ( [ g  /  f ] ta  ->  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
28273ad2ant3 978 . . . . . . . . 9  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  dom  g  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
2926, 28eqtr4d 2331 . . . . . . . 8  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  dom  f  =  dom  g )
30 bnj1322 29171 . . . . . . . . 9  |-  ( dom  f  =  dom  g  ->  ( dom  f  i^i 
dom  g )  =  dom  f )
3130reseq2d 4971 . . . . . . . 8  |-  ( dom  f  =  dom  g  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( f  |`  dom  f ) )
3229, 31syl 15 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( f  |`  dom  f ) )
33 releq 4787 . . . . . . . . 9  |-  ( z  =  f  ->  ( Rel  z  <->  Rel  f ) )
3420, 21, 6bnj66 29208 . . . . . . . . 9  |-  ( z  e.  C  ->  Rel  z )
3533, 34vtoclga 2862 . . . . . . . 8  |-  ( f  e.  C  ->  Rel  f )
36 resdm 5009 . . . . . . . 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 2328 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  f )
39 eqeq2 2305 . . . . . . . . . 10  |-  ( dom  f  =  dom  g  ->  ( ( dom  f  i^i  dom  g )  =  dom  f  <->  ( dom  f  i^i  dom  g )  =  dom  g ) )
4030, 39mpbid 201 . . . . . . . . 9  |-  ( dom  f  =  dom  g  ->  ( dom  f  i^i 
dom  g )  =  dom  g )
4140reseq2d 4971 . . . . . . . 8  |-  ( dom  f  =  dom  g  ->  ( g  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  dom  g ) )
4229, 41syl 15 . . . . . . 7  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( g  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  dom  g ) )
4320, 21, 6bnj66 29208 . . . . . . . 8  |-  ( g  e.  C  ->  Rel  g )
44 resdm 5009 . . . . . . . 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 2328 . . . . . 6  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  ( g  |`  ( dom  f  i^i  dom  g
) )  =  g )
4724, 38, 463eqtr3d 2336 . . . . 5  |-  ( ( R  FrSe  A  /\  ta  /\  [ g  / 
f ] ta )  ->  f  =  g )
48473expib 1154 . . . 4  |-  ( R 
FrSe  A  ->  ( ( ta  /\  [ g  /  f ] ta )  ->  f  =  g ) )
4948alrimivv 1622 . . 3  |-  ( R 
FrSe  A  ->  A. f A. g ( ( ta 
/\  [ g  / 
f ] ta )  ->  f  =  g ) )
5049adantr 451 . 2  |-  ( ( R  FrSe  A  /\  E. f ta )  ->  A. f A. g ( ( ta  /\  [
g  /  f ] ta )  ->  f  =  g ) )
51 nfv 1609 . . 3  |-  F/ g ta
5251eu2 2181 . 2  |-  ( E! f ta  <->  ( E. f ta  /\  A. f A. g ( ( ta 
/\  [ g  / 
f ] ta )  ->  f  =  g ) ) )
531, 50, 52sylanbrc 645 1  |-  ( ( R  FrSe  A  /\  E. f ta )  ->  E! f ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1530   E.wex 1531    = wceq 1632   [wsb 1638    e. wcel 1696   E!weu 2156   {cab 2282   A.wral 2556   E.wrex 2557    u. cun 3163    i^i cin 3164    C_ wss 3165   {csn 3653   <.cop 3656   dom cdm 4705    |` cres 4707   Rel wrel 4710    Fn wfn 5266   ` cfv 5271    predc-bnj14 29029    FrSe w-bnj15 29033    trClc-bnj18 29035
This theorem is referenced by:  bnj1489  29402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-bnj17 29028  df-bnj14 29030  df-bnj13 29032  df-bnj15 29034  df-bnj18 29036  df-bnj19 29038
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