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Theorem bnj1245 29101
Description: Technical lemma for bnj60 29149. 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 28898 . . 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 29100 . . 3  |-  C  =  K
82, 7syl6eleq 2502 . 2  |-  ( ph  ->  h  e.  K )
96abeq2i 2519 . . . . . 6  |-  ( h  e.  K  <->  E. d  e.  B  ( h  Fn  d  /\  A. x  e.  d  ( h `  x )  =  ( G `  Z ) ) )
109bnj1238 28896 . . . . 5  |-  ( h  e.  K  ->  E. d  e.  B  h  Fn  d )
1110bnj1196 28884 . . . 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 2519 . . . . . 6  |-  ( d  e.  B  <->  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
)
1413simplbi 447 . . . . 5  |-  ( d  e.  B  ->  d  C_  A )
15 fndm 5511 . . . . 5  |-  ( h  Fn  d  ->  dom  h  =  d )
1614, 15bnj1241 28897 . . . 4  |-  ( ( d  e.  B  /\  h  Fn  d )  ->  dom  h  C_  A
)
1711, 16bnj593 28831 . . 3  |-  ( h  e.  K  ->  E. d dom  h  C_  A )
1817bnj937 28860 . 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {cab 2398    =/= wne 2575   A.wral 2674   E.wrex 2675   {crab 2678    i^i cin 3287    C_ wss 3288   <.cop 3785   class class class wbr 4180   dom cdm 4845    |` cres 4847    Fn wfn 5416   ` cfv 5421    /\ w-bnj17 28768    predc-bnj14 28770    FrSe w-bnj15 28774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-res 4857  df-iota 5385  df-fun 5423  df-fn 5424  df-fv 5429  df-bnj17 28769
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