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Theorem bnj1384 29378
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
bnj1384.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1384.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1384.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1384.4  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
bnj1384.5  |-  D  =  { x  e.  A  |  -.  E. f ta }
bnj1384.6  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
bnj1384.7  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
bnj1384.8  |-  ( ta'  <->  [. y  /  x ]. ta )
bnj1384.9  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
bnj1384.10  |-  P  = 
U. H
Assertion
Ref Expression
bnj1384  |-  ( R 
FrSe  A  ->  Fun  P
)
Distinct variable groups:    A, d,
f, x    B, f    y, C    G, d, f    R, d, f, x    y, f, x
Allowed substitution hints:    ps( x, y, f, d)    ch( x, y, f, d)    ta( x, y, f, d)    A( y)    B( x, y, d)    C( x, f, d)    D( x, y, f, d)    P( x, y, f, d)    R( y)    G( x, y)    H( x, y, f, d)    Y( x, y, f, d)    ta'( x, y, f, d)

Proof of Theorem bnj1384
Dummy variables  z 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj1384.1 . . . . 5  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
2 bnj1384.2 . . . . 5  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
3 bnj1384.3 . . . . 5  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
4 bnj1384.4 . . . . 5  |-  ( ta  <->  ( f  e.  C  /\  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
5 bnj1384.5 . . . . 5  |-  D  =  { x  e.  A  |  -.  E. f ta }
6 bnj1384.6 . . . . 5  |-  ( ps  <->  ( R  FrSe  A  /\  D  =/=  (/) ) )
7 bnj1384.7 . . . . 5  |-  ( ch  <->  ( ps  /\  x  e.  D  /\  A. y  e.  D  -.  y R x ) )
8 bnj1384.8 . . . . 5  |-  ( ta'  <->  [. y  /  x ]. ta )
9 bnj1384.9 . . . . 5  |-  H  =  { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
10 bnj1384.10 . . . . 5  |-  P  = 
U. H
111, 2, 3, 4, 8bnj1373 29376 . . . . 5  |-  ( ta'  <->  (
f  e.  C  /\  dom  f  =  ( { y }  u.  trCl ( y ,  A ,  R ) ) ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11bnj1371 29375 . . . 4  |-  ( f  e.  H  ->  Fun  f )
1312rgen 2621 . . 3  |-  A. f  e.  H  Fun  f
14 id 19 . . . . . 6  |-  ( R 
FrSe  A  ->  R  FrSe  A )
151, 2, 3, 4, 5, 6, 7, 8, 9bnj1374 29377 . . . . . 6  |-  ( f  e.  H  ->  f  e.  C )
16 nfab1 2434 . . . . . . . . . 10  |-  F/_ f { f  |  E. y  e.  pred  ( x ,  A ,  R
) ta' }
179, 16nfcxfr 2429 . . . . . . . . 9  |-  F/_ f H
1817nfcri 2426 . . . . . . . 8  |-  F/ f  g  e.  H
19 nfab1 2434 . . . . . . . . . 10  |-  F/_ f { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
203, 19nfcxfr 2429 . . . . . . . . 9  |-  F/_ f C
2120nfcri 2426 . . . . . . . 8  |-  F/ f  g  e.  C
2218, 21nfim 1781 . . . . . . 7  |-  F/ f ( g  e.  H  ->  g  e.  C )
23 eleq1 2356 . . . . . . . 8  |-  ( f  =  g  ->  (
f  e.  H  <->  g  e.  H ) )
24 eleq1 2356 . . . . . . . 8  |-  ( f  =  g  ->  (
f  e.  C  <->  g  e.  C ) )
2523, 24imbi12d 311 . . . . . . 7  |-  ( f  =  g  ->  (
( f  e.  H  ->  f  e.  C )  <-> 
( g  e.  H  ->  g  e.  C ) ) )
2622, 25, 15chvar 1939 . . . . . 6  |-  ( g  e.  H  ->  g  e.  C )
27 eqid 2296 . . . . . . 7  |-  ( dom  f  i^i  dom  g
)  =  ( dom  f  i^i  dom  g
)
281, 2, 3, 27bnj1326 29372 . . . . . 6  |-  ( ( R  FrSe  A  /\  f  e.  C  /\  g  e.  C )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  ( dom  f  i^i  dom  g )
) )
2914, 15, 26, 28syl3an 1224 . . . . 5  |-  ( ( R  FrSe  A  /\  f  e.  H  /\  g  e.  H )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  ( dom  f  i^i  dom  g )
) )
30293expib 1154 . . . 4  |-  ( R 
FrSe  A  ->  ( ( f  e.  H  /\  g  e.  H )  ->  ( f  |`  ( dom  f  i^i  dom  g
) )  =  ( g  |`  ( dom  f  i^i  dom  g )
) ) )
3130ralrimivv 2647 . . 3  |-  ( R 
FrSe  A  ->  A. f  e.  H  A. g  e.  H  ( f  |`  ( dom  f  i^i 
dom  g ) )  =  ( g  |`  ( dom  f  i^i  dom  g ) ) )
32 biid 227 . . . 4  |-  ( A. f  e.  H  Fun  f 
<-> 
A. f  e.  H  Fun  f )
33 biid 227 . . . 4  |-  ( ( A. f  e.  H  Fun  f  /\  A. f  e.  H  A. g  e.  H  ( f  |`  ( dom  f  i^i 
dom  g ) )  =  ( g  |`  ( dom  f  i^i  dom  g ) ) )  <-> 
( A. f  e.  H  Fun  f  /\  A. f  e.  H  A. g  e.  H  (
f  |`  ( dom  f  i^i  dom  g ) )  =  ( g  |`  ( dom  f  i^i  dom  g ) ) ) )
349bnj1317 29170 . . . 4  |-  ( z  e.  H  ->  A. f 
z  e.  H )
3532, 27, 33, 34bnj1386 29182 . . 3  |-  ( ( A. f  e.  H  Fun  f  /\  A. f  e.  H  A. g  e.  H  ( f  |`  ( dom  f  i^i 
dom  g ) )  =  ( g  |`  ( dom  f  i^i  dom  g ) ) )  ->  Fun  U. H )
3613, 31, 35sylancr 644 . 2  |-  ( R 
FrSe  A  ->  Fun  U. H )
3710funeqi 5291 . 2  |-  ( Fun 
P  <->  Fun  U. H )
3836, 37sylibr 203 1  |-  ( R 
FrSe  A  ->  Fun  P
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560   [.wsbc 3004    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   <.cop 3656   U.cuni 3843   class class class wbr 4039   dom cdm 4705    |` cres 4707   Fun wfun 5265    Fn wfn 5266   ` cfv 5271    predc-bnj14 29029    FrSe w-bnj15 29033    trClc-bnj18 29035
This theorem is referenced by:  bnj1312  29404
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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