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Theorem fpwwe2lem11 8262
Description: Lemma for fpwwe2 8265. (Contributed by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
fpwwe2.1  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. (
u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }
fpwwe2.2  |-  ( ph  ->  A  e.  _V )
fpwwe2.3  |-  ( (
ph  /\  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) )  -> 
( x F r )  e.  A )
fpwwe2.4  |-  X  = 
U. dom  W
Assertion
Ref Expression
fpwwe2lem11  |-  ( ph  ->  W : dom  W --> ~P ( X  X.  X
) )
Distinct variable groups:    y, u, r, x, F    X, r, u, x, y    ph, r, u, x, y    A, r, x    W, r, u, x, y
Allowed substitution hints:    A( y, u)

Proof of Theorem fpwwe2lem11
Dummy variables  s 
t  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fpwwe2.1 . . . . . 6  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. (
u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }
21relopabi 4811 . . . . 5  |-  Rel  W
32a1i 10 . . . 4  |-  ( ph  ->  Rel  W )
4 simprr 733 . . . . . . . . 9  |-  ( ( ( ph  /\  (
w W s  /\  w W t ) )  /\  ( w  C_  w  /\  s  =  ( t  i^i  ( w  X.  w ) ) ) )  ->  s  =  ( t  i^i  ( w  X.  w
) ) )
5 fpwwe2.2 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  e.  _V )
61, 5fpwwe2lem2 8254 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( w W t  <-> 
( ( w  C_  A  /\  t  C_  (
w  X.  w ) )  /\  ( t  We  w  /\  A. y  e.  w  [. ( `' t " {
y } )  /  u ]. ( u F ( t  i^i  (
u  X.  u ) ) )  =  y ) ) ) )
76simprbda 606 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w W
t )  ->  (
w  C_  A  /\  t  C_  ( w  X.  w ) ) )
87simprd 449 . . . . . . . . . . . 12  |-  ( (
ph  /\  w W
t )  ->  t  C_  ( w  X.  w
) )
98adantrl 696 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  t  C_  (
w  X.  w ) )
109adantr 451 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
w W s  /\  w W t ) )  /\  ( w  C_  w  /\  s  =  ( t  i^i  ( w  X.  w ) ) ) )  ->  t  C_  ( w  X.  w
) )
11 df-ss 3166 . . . . . . . . . 10  |-  ( t 
C_  ( w  X.  w )  <->  ( t  i^i  ( w  X.  w
) )  =  t )
1210, 11sylib 188 . . . . . . . . 9  |-  ( ( ( ph  /\  (
w W s  /\  w W t ) )  /\  ( w  C_  w  /\  s  =  ( t  i^i  ( w  X.  w ) ) ) )  ->  (
t  i^i  ( w  X.  w ) )  =  t )
134, 12eqtrd 2315 . . . . . . . 8  |-  ( ( ( ph  /\  (
w W s  /\  w W t ) )  /\  ( w  C_  w  /\  s  =  ( t  i^i  ( w  X.  w ) ) ) )  ->  s  =  t )
14 simprr 733 . . . . . . . . 9  |-  ( ( ( ph  /\  (
w W s  /\  w W t ) )  /\  ( w  C_  w  /\  t  =  ( s  i^i  ( w  X.  w ) ) ) )  ->  t  =  ( s  i^i  ( w  X.  w
) ) )
151, 5fpwwe2lem2 8254 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( w W s  <-> 
( ( w  C_  A  /\  s  C_  (
w  X.  w ) )  /\  ( s  We  w  /\  A. y  e.  w  [. ( `' s " {
y } )  /  u ]. ( u F ( s  i^i  (
u  X.  u ) ) )  =  y ) ) ) )
1615simprbda 606 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w W
s )  ->  (
w  C_  A  /\  s  C_  ( w  X.  w ) ) )
1716simprd 449 . . . . . . . . . . . 12  |-  ( (
ph  /\  w W
s )  ->  s  C_  ( w  X.  w
) )
1817adantrr 697 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  s  C_  (
w  X.  w ) )
1918adantr 451 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
w W s  /\  w W t ) )  /\  ( w  C_  w  /\  t  =  ( s  i^i  ( w  X.  w ) ) ) )  ->  s  C_  ( w  X.  w
) )
20 df-ss 3166 . . . . . . . . . 10  |-  ( s 
C_  ( w  X.  w )  <->  ( s  i^i  ( w  X.  w
) )  =  s )
2119, 20sylib 188 . . . . . . . . 9  |-  ( ( ( ph  /\  (
w W s  /\  w W t ) )  /\  ( w  C_  w  /\  t  =  ( s  i^i  ( w  X.  w ) ) ) )  ->  (
s  i^i  ( w  X.  w ) )  =  s )
2214, 21eqtr2d 2316 . . . . . . . 8  |-  ( ( ( ph  /\  (
w W s  /\  w W t ) )  /\  ( w  C_  w  /\  t  =  ( s  i^i  ( w  X.  w ) ) ) )  ->  s  =  t )
235adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  A  e.  _V )
24 fpwwe2.3 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) )  -> 
( x F r )  e.  A )
2524adantlr 695 . . . . . . . . 9  |-  ( ( ( ph  /\  (
w W s  /\  w W t ) )  /\  ( x  C_  A  /\  r  C_  (
x  X.  x )  /\  r  We  x
) )  ->  (
x F r )  e.  A )
26 simprl 732 . . . . . . . . 9  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  w W s )
27 simprr 733 . . . . . . . . 9  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  w W t )
281, 23, 25, 26, 27fpwwe2lem10 8261 . . . . . . . 8  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  ( ( w 
C_  w  /\  s  =  ( t  i^i  ( w  X.  w
) ) )  \/  ( w  C_  w  /\  t  =  (
s  i^i  ( w  X.  w ) ) ) ) )
2913, 22, 28mpjaodan 761 . . . . . . 7  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  s  =  t )
3029ex 423 . . . . . 6  |-  ( ph  ->  ( ( w W s  /\  w W t )  ->  s  =  t ) )
3130alrimiv 1617 . . . . 5  |-  ( ph  ->  A. t ( ( w W s  /\  w W t )  -> 
s  =  t ) )
3231alrimivv 1618 . . . 4  |-  ( ph  ->  A. w A. s A. t ( ( w W s  /\  w W t )  -> 
s  =  t ) )
33 dffun2 5265 . . . 4  |-  ( Fun 
W  <->  ( Rel  W  /\  A. w A. s A. t ( ( w W s  /\  w W t )  -> 
s  =  t ) ) )
343, 32, 33sylanbrc 645 . . 3  |-  ( ph  ->  Fun  W )
35 funfn 5283 . . 3  |-  ( Fun 
W  <->  W  Fn  dom  W )
3634, 35sylib 188 . 2  |-  ( ph  ->  W  Fn  dom  W
)
37 vex 2791 . . . . 5  |-  s  e. 
_V
3837elrn 4919 . . . 4  |-  ( s  e.  ran  W  <->  E. w  w W s )
392releldmi 4915 . . . . . . . . . . . 12  |-  ( w W s  ->  w  e.  dom  W )
4039adantl 452 . . . . . . . . . . 11  |-  ( (
ph  /\  w W
s )  ->  w  e.  dom  W )
41 elssuni 3855 . . . . . . . . . . 11  |-  ( w  e.  dom  W  ->  w  C_  U. dom  W
)
4240, 41syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  w W
s )  ->  w  C_ 
U. dom  W )
43 fpwwe2.4 . . . . . . . . . 10  |-  X  = 
U. dom  W
4442, 43syl6sseqr 3225 . . . . . . . . 9  |-  ( (
ph  /\  w W
s )  ->  w  C_  X )
45 xpss12 4792 . . . . . . . . 9  |-  ( ( w  C_  X  /\  w  C_  X )  -> 
( w  X.  w
)  C_  ( X  X.  X ) )
4644, 44, 45syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  w W
s )  ->  (
w  X.  w ) 
C_  ( X  X.  X ) )
4717, 46sstrd 3189 . . . . . . 7  |-  ( (
ph  /\  w W
s )  ->  s  C_  ( X  X.  X
) )
4847ex 423 . . . . . 6  |-  ( ph  ->  ( w W s  ->  s  C_  ( X  X.  X ) ) )
4937elpw 3631 . . . . . 6  |-  ( s  e.  ~P ( X  X.  X )  <->  s  C_  ( X  X.  X
) )
5048, 49syl6ibr 218 . . . . 5  |-  ( ph  ->  ( w W s  ->  s  e.  ~P ( X  X.  X
) ) )
5150exlimdv 1664 . . . 4  |-  ( ph  ->  ( E. w  w W s  ->  s  e.  ~P ( X  X.  X ) ) )
5238, 51syl5bi 208 . . 3  |-  ( ph  ->  ( s  e.  ran  W  ->  s  e.  ~P ( X  X.  X
) ) )
5352ssrdv 3185 . 2  |-  ( ph  ->  ran  W  C_  ~P ( X  X.  X
) )
54 df-f 5259 . 2  |-  ( W : dom  W --> ~P ( X  X.  X )  <->  ( W  Fn  dom  W  /\  ran  W 
C_  ~P ( X  X.  X ) ) )
5536, 53, 54sylanbrc 645 1  |-  ( ph  ->  W : dom  W --> ~P ( X  X.  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   [.wsbc 2991    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   {csn 3640   U.cuni 3827   class class class wbr 4023   {copab 4076    We wwe 4351    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   Rel wrel 4694   Fun wfun 5249    Fn wfn 5250   -->wf 5251  (class class class)co 5858
This theorem is referenced by:  fpwwe2lem13  8264  fpwwe2  8265
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-riota 6304  df-recs 6388  df-oi 7225
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