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Theorem bnj1245 29456
Description: Technical lemma for bnj60 29504. 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
bnj1245.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1245.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1245.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1245.4  |-  D  =  ( dom  g  i^i 
dom  h )
bnj1245.5  |-  E  =  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
bnj1245.6  |-  ( ph  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
bnj1245.7  |-  ( ps  <->  (
ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )
bnj1245.8  |-  Z  = 
<. x ,  ( h  |`  pred ( x ,  A ,  R ) ) >.
bnj1245.9  |-  K  =  { h  |  E. d  e.  B  (
h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  Z ) ) }
Assertion
Ref Expression
bnj1245  |-  ( ph  ->  dom  h  C_  A
)
Distinct variable groups:    A, d    B, f, h    f, G, h    h, Y    f, Z    f, d, h    x, f, h
Allowed substitution hints:    ph( x, y, f, g, h, d)    ps( x, y, f, g, h, d)    A( x, y, f, g, h)    B( x, y, g, d)    C( x, y, f, g, h, d)    D( x, y, f, g, h, d)    R( x, y, f, g, h, d)    E( x, y, f, g, h, d)    G( x, y, g, d)    K( x, y, f, g, h, d)    Y( x, y, f, g, d)    Z( x, y, g, h, d)

Proof of Theorem bnj1245
StepHypRef Expression
1 bnj1245.6 . . . 4  |-  ( ph  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
21bnj1247 29253 . . 3  |-  ( ph  ->  h  e.  C )
3 bnj1245.2 . . . 4  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
4 bnj1245.3 . . . 4  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
5 bnj1245.8 . . . 4  |-  Z  = 
<. x ,  ( h  |`  pred ( x ,  A ,  R ) ) >.
6 bnj1245.9 . . . 4  |-  K  =  { h  |  E. d  e.  B  (
h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  Z ) ) }
73, 4, 5, 6bnj1234 29455 . . 3  |-  C  =  K
82, 7syl6eleq 2528 . 2  |-  ( ph  ->  h  e.  K )
96abeq2i 2545 . . . . . 6  |-  ( h  e.  K  <->  E. d  e.  B  ( h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  Z ) ) )
109bnj1238 29251 . . . . 5  |-  ( h  e.  K  ->  E. d  e.  B  h  Fn  d )
1110bnj1196 29239 . . . 4  |-  ( h  e.  K  ->  E. d
( d  e.  B  /\  h  Fn  d
) )
12 bnj1245.1 . . . . . . 7  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
1312abeq2i 2545 . . . . . 6  |-  ( d  e.  B  <->  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
)
1413simplbi 448 . . . . 5  |-  ( d  e.  B  ->  d  C_  A )
15 fndm 5547 . . . . 5  |-  ( h  Fn  d  ->  dom  h  =  d )
1614, 15bnj1241 29252 . . . 4  |-  ( ( d  e.  B  /\  h  Fn  d )  ->  dom  h  C_  A
)
1711, 16bnj593 29186 . . 3  |-  ( h  e.  K  ->  E. d dom  h  C_  A )
1817bnj937 29215 . 2  |-  ( h  e.  K  ->  dom  h  C_  A )
198, 18syl 16 1  |-  ( ph  ->  dom  h  C_  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {cab 2424    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711    i^i cin 3321    C_ wss 3322   <.cop 3819   class class class wbr 4215   dom cdm 4881    |` cres 4883    Fn wfn 5452   ` cfv 5457    /\ w-bnj17 29123    predc-bnj14 29125    FrSe w-bnj15 29129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-res 4893  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465  df-bnj17 29124
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