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Theorem bnj1525 29099
Description: Technical lemma for bnj1522 29102. 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
bnj1525.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1525.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1525.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1525.4  |-  F  = 
U. C
bnj1525.5  |-  ( ph  <->  ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) ) )
bnj1525.6  |-  ( ps  <->  (
ph  /\  F  =/=  H ) )
Assertion
Ref Expression
bnj1525  |-  ( ps 
->  A. x ps )
Distinct variable groups:    x, A    x, H    x, R    x, d    x, f
Allowed substitution hints:    ph( x, f, d)    ps( x, f, d)    A( f, d)    B( x, f, d)    C( x, f, d)    R( f, d)    F( x, f, d)    G( x, f, d)    H( f, d)    Y( x, f, d)

Proof of Theorem bnj1525
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 bnj1525.6 . . 3  |-  ( ps  <->  (
ph  /\  F  =/=  H ) )
2 bnj1525.5 . . . . 5  |-  ( ph  <->  ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) ) )
3 nfv 1605 . . . . . 6  |-  F/ x  R  FrSe  A
4 nfv 1605 . . . . . 6  |-  F/ x  H  Fn  A
5 nfra1 2593 . . . . . 6  |-  F/ x A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. )
63, 4, 5nf3an 1774 . . . . 5  |-  F/ x
( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )
72, 6nfxfr 1557 . . . 4  |-  F/ x ph
8 bnj1525.4 . . . . . 6  |-  F  = 
U. C
9 bnj1525.3 . . . . . . . . 9  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
10 bnj1525.1 . . . . . . . . . 10  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
1110bnj1309 29052 . . . . . . . . 9  |-  ( w  e.  B  ->  A. x  w  e.  B )
129, 11bnj1307 29053 . . . . . . . 8  |-  ( w  e.  C  ->  A. x  w  e.  C )
1312nfcii 2410 . . . . . . 7  |-  F/_ x C
1413nfuni 3833 . . . . . 6  |-  F/_ x U. C
158, 14nfcxfr 2416 . . . . 5  |-  F/_ x F
16 nfcv 2419 . . . . 5  |-  F/_ x H
1715, 16nfne 2539 . . . 4  |-  F/ x  F  =/=  H
187, 17nfan 1771 . . 3  |-  F/ x
( ph  /\  F  =/= 
H )
191, 18nfxfr 1557 . 2  |-  F/ x ps
2019nfri 1742 1  |-  ( ps 
->  A. x ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527    = wceq 1623   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   <.cop 3643   U.cuni 3827    |` cres 4691    Fn wfn 5250   ` cfv 5255    predc-bnj14 28713    FrSe w-bnj15 28717
This theorem is referenced by:  bnj1523  29101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-uni 3828
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