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Theorem fpwwe2lem11 8515
Description: Lemma for fpwwe2 8518. (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 5000 . . . . 5  |-  Rel  W
32a1i 11 . . . 4  |-  ( ph  ->  Rel  W )
4 simprr 734 . . . . . . . . 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 8507 . . . . . . . . . . . . . 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 607 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w W
t )  ->  (
w  C_  A  /\  t  C_  ( w  X.  w ) ) )
87simprd 450 . . . . . . . . . . . 12  |-  ( (
ph  /\  w W
t )  ->  t  C_  ( w  X.  w
) )
98adantrl 697 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  t  C_  (
w  X.  w ) )
109adantr 452 . . . . . . . . . 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 3334 . . . . . . . . . 10  |-  ( t 
C_  ( w  X.  w )  <->  ( t  i^i  ( w  X.  w
) )  =  t )
1210, 11sylib 189 . . . . . . . . 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 2468 . . . . . . . 8  |-  ( ( ( ph  /\  (
w W s  /\  w W t ) )  /\  ( w  C_  w  /\  s  =  ( t  i^i  ( w  X.  w ) ) ) )  ->  s  =  t )
14 simprr 734 . . . . . . . . 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 8507 . . . . . . . . . . . . . 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 607 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w W
s )  ->  (
w  C_  A  /\  s  C_  ( w  X.  w ) ) )
1716simprd 450 . . . . . . . . . . . 12  |-  ( (
ph  /\  w W
s )  ->  s  C_  ( w  X.  w
) )
1817adantrr 698 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  s  C_  (
w  X.  w ) )
1918adantr 452 . . . . . . . . . 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 3334 . . . . . . . . . 10  |-  ( s 
C_  ( w  X.  w )  <->  ( s  i^i  ( w  X.  w
) )  =  s )
2119, 20sylib 189 . . . . . . . . 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 2469 . . . . . . . 8  |-  ( ( ( ph  /\  (
w W s  /\  w W t ) )  /\  ( w  C_  w  /\  t  =  ( s  i^i  ( w  X.  w ) ) ) )  ->  s  =  t )
235adantr 452 . . . . . . . . 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 696 . . . . . . . . 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 733 . . . . . . . . 9  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  w W s )
27 simprr 734 . . . . . . . . 9  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  w W t )
281, 23, 25, 26, 27fpwwe2lem10 8514 . . . . . . . 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 762 . . . . . . 7  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  s  =  t )
3029ex 424 . . . . . 6  |-  ( ph  ->  ( ( w W s  /\  w W t )  ->  s  =  t ) )
3130alrimiv 1641 . . . . 5  |-  ( ph  ->  A. t ( ( w W s  /\  w W t )  -> 
s  =  t ) )
3231alrimivv 1642 . . . 4  |-  ( ph  ->  A. w A. s A. t ( ( w W s  /\  w W t )  -> 
s  =  t ) )
33 dffun2 5464 . . . 4  |-  ( Fun 
W  <->  ( Rel  W  /\  A. w A. s A. t ( ( w W s  /\  w W t )  -> 
s  =  t ) ) )
343, 32, 33sylanbrc 646 . . 3  |-  ( ph  ->  Fun  W )
35 funfn 5482 . . 3  |-  ( Fun 
W  <->  W  Fn  dom  W )
3634, 35sylib 189 . 2  |-  ( ph  ->  W  Fn  dom  W
)
37 vex 2959 . . . . 5  |-  s  e. 
_V
3837elrn 5110 . . . 4  |-  ( s  e.  ran  W  <->  E. w  w W s )
392releldmi 5106 . . . . . . . . . . . 12  |-  ( w W s  ->  w  e.  dom  W )
4039adantl 453 . . . . . . . . . . 11  |-  ( (
ph  /\  w W
s )  ->  w  e.  dom  W )
41 elssuni 4043 . . . . . . . . . . 11  |-  ( w  e.  dom  W  ->  w  C_  U. dom  W
)
4240, 41syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  w W
s )  ->  w  C_ 
U. dom  W )
43 fpwwe2.4 . . . . . . . . . 10  |-  X  = 
U. dom  W
4442, 43syl6sseqr 3395 . . . . . . . . 9  |-  ( (
ph  /\  w W
s )  ->  w  C_  X )
45 xpss12 4981 . . . . . . . . 9  |-  ( ( w  C_  X  /\  w  C_  X )  -> 
( w  X.  w
)  C_  ( X  X.  X ) )
4644, 44, 45syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  w W
s )  ->  (
w  X.  w ) 
C_  ( X  X.  X ) )
4717, 46sstrd 3358 . . . . . . 7  |-  ( (
ph  /\  w W
s )  ->  s  C_  ( X  X.  X
) )
4847ex 424 . . . . . 6  |-  ( ph  ->  ( w W s  ->  s  C_  ( X  X.  X ) ) )
4937elpw 3805 . . . . . 6  |-  ( s  e.  ~P ( X  X.  X )  <->  s  C_  ( X  X.  X
) )
5048, 49syl6ibr 219 . . . . 5  |-  ( ph  ->  ( w W s  ->  s  e.  ~P ( X  X.  X
) ) )
5150exlimdv 1646 . . . 4  |-  ( ph  ->  ( E. w  w W s  ->  s  e.  ~P ( X  X.  X ) ) )
5238, 51syl5bi 209 . . 3  |-  ( ph  ->  ( s  e.  ran  W  ->  s  e.  ~P ( X  X.  X
) ) )
5352ssrdv 3354 . 2  |-  ( ph  ->  ran  W  C_  ~P ( X  X.  X
) )
54 df-f 5458 . 2  |-  ( W : dom  W --> ~P ( X  X.  X )  <->  ( W  Fn  dom  W  /\  ran  W 
C_  ~P ( X  X.  X ) ) )
5536, 53, 54sylanbrc 646 1  |-  ( ph  ->  W : dom  W --> ~P ( X  X.  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956   [.wsbc 3161    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   {csn 3814   U.cuni 4015   class class class wbr 4212   {copab 4265    We wwe 4540    X. cxp 4876   `'ccnv 4877   dom cdm 4878   ran crn 4879   "cima 4881   Rel wrel 4883   Fun wfun 5448    Fn wfn 5449   -->wf 5450  (class class class)co 6081
This theorem is referenced by:  fpwwe2lem13  8517  fpwwe2  8518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-riota 6549  df-recs 6633  df-oi 7479
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