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Theorem bnj1286 28726
Description: Technical lemma for bnj60 28769. 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 28722 . . . . . . . 8  |-  ( ph  ->  E. d  e.  B  g  Fn  d )
98bnj1196 28504 . . . . . . 7  |-  ( ph  ->  E. d ( d  e.  B  /\  g  Fn  d ) )
102bnj1517 28559 . . . . . . . . 9  |-  ( d  e.  B  ->  A. x  e.  d  pred ( x ,  A ,  R
)  C_  d )
1110adantr 452 . . . . . . . 8  |-  ( ( d  e.  B  /\  g  Fn  d )  ->  A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d )
12 fndm 5484 . . . . . . . . . 10  |-  ( g  Fn  d  ->  dom  g  =  d )
13 sseq2 3313 . . . . . . . . . . 11  |-  ( dom  g  =  d  -> 
(  pred ( x ,  A ,  R ) 
C_  dom  g  <->  pred ( x ,  A ,  R
)  C_  d )
)
1413raleqbi1dv 2855 . . . . . . . . . 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 16 . . . . . . . . 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 453 . . . . . . . 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 224 . . . . . . 7  |-  ( ( d  e.  B  /\  g  Fn  d )  ->  A. x  e.  dom  g  pred ( x ,  A ,  R ) 
C_  dom  g )
189, 17bnj593 28451 . . . . . 6  |-  ( ph  ->  E. d A. x  e.  dom  g  pred (
x ,  A ,  R )  C_  dom  g )
1918bnj937 28480 . . . . 5  |-  ( ph  ->  A. x  e.  dom  g  pred ( x ,  A ,  R ) 
C_  dom  g )
201, 19bnj835 28466 . . . 4  |-  ( ps 
->  A. x  e.  dom  g  pred ( x ,  A ,  R ) 
C_  dom  g )
216bnj21 28420 . . . . . . 7  |-  E  C_  D
225bnj1292 28525 . . . . . . 7  |-  D  C_  dom  g
2321, 22sstri 3300 . . . . . 6  |-  E  C_  dom  g
2423sseli 3287 . . . . 5  |-  ( x  e.  E  ->  x  e.  dom  g )
251, 24bnj836 28467 . . . 4  |-  ( ps 
->  x  e.  dom  g )
2620, 25bnj1294 28527 . . 3  |-  ( ps 
->  pred ( x ,  A ,  R ) 
C_  dom  g )
272, 3, 4, 5, 6, 7, 1bnj1259 28723 . . . . . . . 8  |-  ( ph  ->  E. d  e.  B  h  Fn  d )
2827bnj1196 28504 . . . . . . 7  |-  ( ph  ->  E. d ( d  e.  B  /\  h  Fn  d ) )
2910adantr 452 . . . . . . . 8  |-  ( ( d  e.  B  /\  h  Fn  d )  ->  A. x  e.  d 
pred ( x ,  A ,  R ) 
C_  d )
30 fndm 5484 . . . . . . . . . 10  |-  ( h  Fn  d  ->  dom  h  =  d )
31 sseq2 3313 . . . . . . . . . . 11  |-  ( dom  h  =  d  -> 
(  pred ( x ,  A ,  R ) 
C_  dom  h  <->  pred ( x ,  A ,  R
)  C_  d )
)
3231raleqbi1dv 2855 . . . . . . . . . 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 16 . . . . . . . . 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 453 . . . . . . . 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 224 . . . . . . 7  |-  ( ( d  e.  B  /\  h  Fn  d )  ->  A. x  e.  dom  h  pred ( x ,  A ,  R ) 
C_  dom  h )
3628, 35bnj593 28451 . . . . . 6  |-  ( ph  ->  E. d A. x  e.  dom  h  pred (
x ,  A ,  R )  C_  dom  h )
3736bnj937 28480 . . . . 5  |-  ( ph  ->  A. x  e.  dom  h  pred ( x ,  A ,  R ) 
C_  dom  h )
381, 37bnj835 28466 . . . 4  |-  ( ps 
->  A. x  e.  dom  h  pred ( x ,  A ,  R ) 
C_  dom  h )
395bnj1293 28526 . . . . . . 7  |-  D  C_  dom  h
4021, 39sstri 3300 . . . . . 6  |-  E  C_  dom  h
4140sseli 3287 . . . . 5  |-  ( x  e.  E  ->  x  e.  dom  h )
421, 41bnj836 28467 . . . 4  |-  ( ps 
->  x  e.  dom  h )
4338, 42bnj1294 28527 . . 3  |-  ( ps 
->  pred ( x ,  A ,  R ) 
C_  dom  h )
4426, 43ssind 3508 . 2  |-  ( ps 
->  pred ( x ,  A ,  R ) 
C_  ( dom  g  i^i  dom  h ) )
4544, 5syl6sseqr 3338 1  |-  ( ps 
->  pred ( x ,  A ,  R ) 
C_  D )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {cab 2373    =/= wne 2550   A.wral 2649   E.wrex 2650   {crab 2653    i^i cin 3262    C_ wss 3263   <.cop 3760   class class class wbr 4153   dom cdm 4818    |` cres 4820    Fn wfn 5389   ` cfv 5394    /\ w-bnj17 28388    predc-bnj14 28390    FrSe w-bnj15 28394
This theorem is referenced by:  bnj1280  28727
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
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 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-res 4830  df-iota 5358  df-fun 5396  df-fn 5397  df-fv 5402  df-bnj17 28389
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