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Theorem bnj1245 28789
Description: Technical lemma for bnj60 28837. 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 28586 . . 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 28788 . . 3  |-  C  =  K
82, 7syl6eleq 2448 . 2  |-  ( ph  ->  h  e.  K )
96abeq2i 2465 . . . . . 6  |-  ( h  e.  K  <->  E. d  e.  B  ( h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  Z ) ) )
109bnj1238 28584 . . . . 5  |-  ( h  e.  K  ->  E. d  e.  B  h  Fn  d )
1110bnj1196 28572 . . . 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 2465 . . . . . 6  |-  ( d  e.  B  <->  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
)
1413simplbi 446 . . . . 5  |-  ( d  e.  B  ->  d  C_  A )
15 fndm 5422 . . . . 5  |-  ( h  Fn  d  ->  dom  h  =  d )
1614, 15bnj1241 28585 . . . 4  |-  ( ( d  e.  B  /\  h  Fn  d )  ->  dom  h  C_  A
)
1711, 16bnj593 28519 . . 3  |-  ( h  e.  K  ->  E. d dom  h  C_  A )
1817bnj937 28548 . 2  |-  ( h  e.  K  ->  dom  h  C_  A )
198, 18syl 15 1  |-  ( ph  ->  dom  h  C_  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   {cab 2344    =/= wne 2521   A.wral 2619   E.wrex 2620   {crab 2623    i^i cin 3227    C_ wss 3228   <.cop 3719   class class class wbr 4102   dom cdm 4768    |` cres 4770    Fn wfn 5329   ` cfv 5334    /\ w-bnj17 28456    predc-bnj14 28458    FrSe w-bnj15 28462
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-res 4780  df-iota 5298  df-fun 5336  df-fn 5337  df-fv 5342  df-bnj17 28457
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