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Theorem bnj1286 29365
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
bnj1286.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1286.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1286.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1286.4  |-  D  =  ( dom  g  i^i 
dom  h )
bnj1286.5  |-  E  =  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
bnj1286.6  |-  ( ph  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
bnj1286.7  |-  ( ps  <->  (
ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )
Assertion
Ref Expression
bnj1286  |-  ( ps 
->  pred ( x ,  A ,  R ) 
C_  D )
Distinct variable groups:    A, d,
f    B, f, g    B, h, f    x, D    f, G, g    h, G    R, d, f    g, Y    h, Y    g, d, x, f   
h, d, x
Allowed substitution hints:    ph( x, y, f, g, h, d)    ps( x, y, f, g, h, d)    A( x, y, g, h)    B( x, y, d)    C( x, y, f, g, h, d)    D( y, f, g, h, d)    R( x, y, g, h)    E( x, y, f, g, h, d)    G( x, y, d)    Y( x, y, f, d)

Proof of Theorem bnj1286
StepHypRef Expression
1 bnj1286.7 . . . . 5  |-  ( ps  <->  (
ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )
2 bnj1286.1 . . . . . . . . 9  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
3 bnj1286.2 . . . . . . . . 9  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
4 bnj1286.3 . . . . . . . . 9  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
5 bnj1286.4 . . . . . . . . 9  |-  D  =  ( dom  g  i^i 
dom  h )
6 bnj1286.5 . . . . . . . . 9  |-  E  =  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
7 bnj1286.6 . . . . . . . . 9  |-  ( ph  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
82, 3, 4, 5, 6, 7, 1bnj1256 29361 . . . . . . . 8  |-  ( ph  ->  E. d  e.  B  g  Fn  d )
98bnj1196 29143 . . . . . . 7  |-  ( ph  ->  E. d ( d  e.  B  /\  g  Fn  d ) )
102bnj1517 29198 . . . . . . . . 9  |-  ( d  e.  B  ->  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
1110adantr 451 . . . . . . . 8  |-  ( ( d  e.  B  /\  g  Fn  d )  ->  A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d )
12 fndm 5359 . . . . . . . . . 10  |-  ( g  Fn  d  ->  dom  g  =  d )
13 sseq2 3213 . . . . . . . . . . 11  |-  ( dom  g  =  d  -> 
(  pred ( x ,  A ,  R ) 
C_  dom  g  <->  pred ( x ,  A ,  R
)  C_  d )
)
1413raleqbi1dv 2757 . . . . . . . . . 10  |-  ( dom  g  =  d  -> 
( A. x  e. 
dom  g  pred (
x ,  A ,  R )  C_  dom  g 
<-> 
A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d ) )
1512, 14syl 15 . . . . . . . . 9  |-  ( g  Fn  d  ->  ( A. x  e.  dom  g  pred ( x ,  A ,  R ) 
C_  dom  g  <->  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
)
1615adantl 452 . . . . . . . 8  |-  ( ( d  e.  B  /\  g  Fn  d )  ->  ( A. x  e. 
dom  g  pred (
x ,  A ,  R )  C_  dom  g 
<-> 
A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d ) )
1711, 16mpbird 223 . . . . . . 7  |-  ( ( d  e.  B  /\  g  Fn  d )  ->  A. x  e.  dom  g  pred ( x ,  A ,  R ) 
C_  dom  g )
189, 17bnj593 29090 . . . . . 6  |-  ( ph  ->  E. d A. x  e.  dom  g  pred (
x ,  A ,  R )  C_  dom  g )
1918bnj937 29119 . . . . 5  |-  ( ph  ->  A. x  e.  dom  g  pred ( x ,  A ,  R ) 
C_  dom  g )
201, 19bnj835 29105 . . . 4  |-  ( ps 
->  A. x  e.  dom  g  pred ( x ,  A ,  R ) 
C_  dom  g )
216bnj21 29059 . . . . . . 7  |-  E  C_  D
225bnj1292 29164 . . . . . . 7  |-  D  C_  dom  g
2321, 22sstri 3201 . . . . . 6  |-  E  C_  dom  g
2423sseli 3189 . . . . 5  |-  ( x  e.  E  ->  x  e.  dom  g )
251, 24bnj836 29106 . . . 4  |-  ( ps 
->  x  e.  dom  g )
2620, 25bnj1294 29166 . . 3  |-  ( ps 
->  pred ( x ,  A ,  R ) 
C_  dom  g )
272, 3, 4, 5, 6, 7, 1bnj1259 29362 . . . . . . . 8  |-  ( ph  ->  E. d  e.  B  h  Fn  d )
2827bnj1196 29143 . . . . . . 7  |-  ( ph  ->  E. d ( d  e.  B  /\  h  Fn  d ) )
2910adantr 451 . . . . . . . 8  |-  ( ( d  e.  B  /\  h  Fn  d )  ->  A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d )
30 fndm 5359 . . . . . . . . . 10  |-  ( h  Fn  d  ->  dom  h  =  d )
31 sseq2 3213 . . . . . . . . . . 11  |-  ( dom  h  =  d  -> 
(  pred ( x ,  A ,  R ) 
C_  dom  h  <->  pred ( x ,  A ,  R
)  C_  d )
)
3231raleqbi1dv 2757 . . . . . . . . . 10  |-  ( dom  h  =  d  -> 
( A. x  e. 
dom  h  pred (
x ,  A ,  R )  C_  dom  h 
<-> 
A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d ) )
3330, 32syl 15 . . . . . . . . 9  |-  ( h  Fn  d  ->  ( A. x  e.  dom  h  pred ( x ,  A ,  R ) 
C_  dom  h  <->  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
)
3433adantl 452 . . . . . . . 8  |-  ( ( d  e.  B  /\  h  Fn  d )  ->  ( A. x  e. 
dom  h  pred (
x ,  A ,  R )  C_  dom  h 
<-> 
A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d ) )
3529, 34mpbird 223 . . . . . . 7  |-  ( ( d  e.  B  /\  h  Fn  d )  ->  A. x  e.  dom  h  pred ( x ,  A ,  R ) 
C_  dom  h )
3628, 35bnj593 29090 . . . . . 6  |-  ( ph  ->  E. d A. x  e.  dom  h  pred (
x ,  A ,  R )  C_  dom  h )
3736bnj937 29119 . . . . 5  |-  ( ph  ->  A. x  e.  dom  h  pred ( x ,  A ,  R ) 
C_  dom  h )
381, 37bnj835 29105 . . . 4  |-  ( ps 
->  A. x  e.  dom  h  pred ( x ,  A ,  R ) 
C_  dom  h )
395bnj1293 29165 . . . . . . 7  |-  D  C_  dom  h
4021, 39sstri 3201 . . . . . 6  |-  E  C_  dom  h
4140sseli 3189 . . . . 5  |-  ( x  e.  E  ->  x  e.  dom  h )
421, 41bnj836 29106 . . . 4  |-  ( ps 
->  x  e.  dom  h )
4338, 42bnj1294 29166 . . 3  |-  ( ps 
->  pred ( x ,  A ,  R ) 
C_  dom  h )
4426, 43ssind 3406 . 2  |-  ( ps 
->  pred ( x ,  A ,  R ) 
C_  ( dom  g  i^i  dom  h ) )
4544, 5syl6sseqr 3238 1  |-  ( ps 
->  pred ( x ,  A ,  R ) 
C_  D )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    i^i cin 3164    C_ wss 3165   <.cop 3656   class class class wbr 4039   dom cdm 4705    |` cres 4707    Fn wfn 5266   ` cfv 5271    /\ w-bnj17 29027    predc-bnj14 29029    FrSe w-bnj15 29033
This theorem is referenced by:  bnj1280  29366
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-bnj17 29028
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