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Theorem fpwwe2lem11 9070
Description: Lemma for fpwwe2 9073. (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 4962 . . . . 5  |-  Rel  W
32a1i 11 . . . 4  |-  ( ph  ->  Rel  W )
4 simprr 767 . . . . . . . . 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 9062 . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w W
t )  ->  (
w  C_  A  /\  t  C_  ( w  X.  w ) ) )
87simprd 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  w W
t )  ->  t  C_  ( w  X.  w
) )
98adantrl 723 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  t  C_  (
w  X.  w ) )
109adantr 467 . . . . . . . . . 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 3420 . . . . . . . . . 10  |-  ( t 
C_  ( w  X.  w )  <->  ( t  i^i  ( w  X.  w
) )  =  t )
1210, 11sylib 200 . . . . . . . . 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 2487 . . . . . . . 8  |-  ( ( ( ph  /\  (
w W s  /\  w W t ) )  /\  ( w  C_  w  /\  s  =  ( t  i^i  ( w  X.  w ) ) ) )  ->  s  =  t )
14 simprr 767 . . . . . . . . 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 9062 . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w W
s )  ->  (
w  C_  A  /\  s  C_  ( w  X.  w ) ) )
1716simprd 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  w W
s )  ->  s  C_  ( w  X.  w
) )
1817adantrr 724 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  s  C_  (
w  X.  w ) )
1918adantr 467 . . . . . . . . . 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 3420 . . . . . . . . . 10  |-  ( s 
C_  ( w  X.  w )  <->  ( s  i^i  ( w  X.  w
) )  =  s )
2119, 20sylib 200 . . . . . . . . 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 2488 . . . . . . . 8  |-  ( ( ( ph  /\  (
w W s  /\  w W t ) )  /\  ( w  C_  w  /\  t  =  ( s  i^i  ( w  X.  w ) ) ) )  ->  s  =  t )
235adantr 467 . . . . . . . . 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 722 . . . . . . . . 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 765 . . . . . . . . 9  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  w W s )
27 simprr 767 . . . . . . . . 9  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  w W t )
281, 23, 25, 26, 27fpwwe2lem10 9069 . . . . . . . 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 796 . . . . . . 7  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  s  =  t )
3029ex 436 . . . . . 6  |-  ( ph  ->  ( ( w W s  /\  w W t )  ->  s  =  t ) )
3130alrimiv 1775 . . . . 5  |-  ( ph  ->  A. t ( ( w W s  /\  w W t )  -> 
s  =  t ) )
3231alrimivv 1776 . . . 4  |-  ( ph  ->  A. w A. s A. t ( ( w W s  /\  w W t )  -> 
s  =  t ) )
33 dffun2 5595 . . . 4  |-  ( Fun 
W  <->  ( Rel  W  /\  A. w A. s A. t ( ( w W s  /\  w W t )  -> 
s  =  t ) ) )
343, 32, 33sylanbrc 671 . . 3  |-  ( ph  ->  Fun  W )
35 funfn 5614 . . 3  |-  ( Fun 
W  <->  W  Fn  dom  W )
3634, 35sylib 200 . 2  |-  ( ph  ->  W  Fn  dom  W
)
37 vex 3050 . . . . 5  |-  s  e. 
_V
3837elrn 5078 . . . 4  |-  ( s  e.  ran  W  <->  E. w  w W s )
392releldmi 5074 . . . . . . . . . . . 12  |-  ( w W s  ->  w  e.  dom  W )
4039adantl 468 . . . . . . . . . . 11  |-  ( (
ph  /\  w W
s )  ->  w  e.  dom  W )
41 elssuni 4230 . . . . . . . . . . 11  |-  ( w  e.  dom  W  ->  w  C_  U. dom  W
)
4240, 41syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  w W
s )  ->  w  C_ 
U. dom  W )
43 fpwwe2.4 . . . . . . . . . 10  |-  X  = 
U. dom  W
4442, 43syl6sseqr 3481 . . . . . . . . 9  |-  ( (
ph  /\  w W
s )  ->  w  C_  X )
45 xpss12 4943 . . . . . . . . 9  |-  ( ( w  C_  X  /\  w  C_  X )  -> 
( w  X.  w
)  C_  ( X  X.  X ) )
4644, 44, 45syl2anc 667 . . . . . . . 8  |-  ( (
ph  /\  w W
s )  ->  (
w  X.  w ) 
C_  ( X  X.  X ) )
4717, 46sstrd 3444 . . . . . . 7  |-  ( (
ph  /\  w W
s )  ->  s  C_  ( X  X.  X
) )
4847ex 436 . . . . . 6  |-  ( ph  ->  ( w W s  ->  s  C_  ( X  X.  X ) ) )
49 selpw 3960 . . . . . 6  |-  ( s  e.  ~P ( X  X.  X )  <->  s  C_  ( X  X.  X
) )
5048, 49syl6ibr 231 . . . . 5  |-  ( ph  ->  ( w W s  ->  s  e.  ~P ( X  X.  X
) ) )
5150exlimdv 1781 . . . 4  |-  ( ph  ->  ( E. w  w W s  ->  s  e.  ~P ( X  X.  X ) ) )
5238, 51syl5bi 221 . . 3  |-  ( ph  ->  ( s  e.  ran  W  ->  s  e.  ~P ( X  X.  X
) ) )
5352ssrdv 3440 . 2  |-  ( ph  ->  ran  W  C_  ~P ( X  X.  X
) )
54 df-f 5589 . 2  |-  ( W : dom  W --> ~P ( X  X.  X )  <->  ( W  Fn  dom  W  /\  ran  W 
C_  ~P ( X  X.  X ) ) )
5536, 53, 54sylanbrc 671 1  |-  ( ph  ->  W : dom  W --> ~P ( X  X.  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 986   A.wal 1444    = wceq 1446   E.wex 1665    e. wcel 1889   A.wral 2739   _Vcvv 3047   [.wsbc 3269    i^i cin 3405    C_ wss 3406   ~Pcpw 3953   {csn 3970   U.cuni 4201   class class class wbr 4405   {copab 4463    We wwe 4795    X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840   Rel wrel 4842   Fun wfun 5579    Fn wfn 5580   -->wf 5581  (class class class)co 6295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-wrecs 7033  df-recs 7095  df-oi 8030
This theorem is referenced by:  fpwwe2lem13  9072  fpwwe2  9073
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