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Theorem fpwwe2lem11 8799
Description: Lemma for fpwwe2 8802. (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 4960 . . . . 5  |-  Rel  W
32a1i 11 . . . 4  |-  ( ph  ->  Rel  W )
4 simprr 756 . . . . . . . . 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 8791 . . . . . . . . . . . . . 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 623 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w W
t )  ->  (
w  C_  A  /\  t  C_  ( w  X.  w ) ) )
87simprd 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  w W
t )  ->  t  C_  ( w  X.  w
) )
98adantrl 715 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  t  C_  (
w  X.  w ) )
109adantr 465 . . . . . . . . . 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 3337 . . . . . . . . . 10  |-  ( t 
C_  ( w  X.  w )  <->  ( t  i^i  ( w  X.  w
) )  =  t )
1210, 11sylib 196 . . . . . . . . 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 2470 . . . . . . . 8  |-  ( ( ( ph  /\  (
w W s  /\  w W t ) )  /\  ( w  C_  w  /\  s  =  ( t  i^i  ( w  X.  w ) ) ) )  ->  s  =  t )
14 simprr 756 . . . . . . . . 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 8791 . . . . . . . . . . . . . 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 623 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w W
s )  ->  (
w  C_  A  /\  s  C_  ( w  X.  w ) ) )
1716simprd 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  w W
s )  ->  s  C_  ( w  X.  w
) )
1817adantrr 716 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  s  C_  (
w  X.  w ) )
1918adantr 465 . . . . . . . . . 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 3337 . . . . . . . . . 10  |-  ( s 
C_  ( w  X.  w )  <->  ( s  i^i  ( w  X.  w
) )  =  s )
2119, 20sylib 196 . . . . . . . . 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 2471 . . . . . . . 8  |-  ( ( ( ph  /\  (
w W s  /\  w W t ) )  /\  ( w  C_  w  /\  t  =  ( s  i^i  ( w  X.  w ) ) ) )  ->  s  =  t )
235adantr 465 . . . . . . . . 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 714 . . . . . . . . 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 755 . . . . . . . . 9  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  w W s )
27 simprr 756 . . . . . . . . 9  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  w W t )
281, 23, 25, 26, 27fpwwe2lem10 8798 . . . . . . . 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 784 . . . . . . 7  |-  ( (
ph  /\  ( w W s  /\  w W t ) )  ->  s  =  t )
3029ex 434 . . . . . 6  |-  ( ph  ->  ( ( w W s  /\  w W t )  ->  s  =  t ) )
3130alrimiv 1685 . . . . 5  |-  ( ph  ->  A. t ( ( w W s  /\  w W t )  -> 
s  =  t ) )
3231alrimivv 1686 . . . 4  |-  ( ph  ->  A. w A. s A. t ( ( w W s  /\  w W t )  -> 
s  =  t ) )
33 dffun2 5423 . . . 4  |-  ( Fun 
W  <->  ( Rel  W  /\  A. w A. s A. t ( ( w W s  /\  w W t )  -> 
s  =  t ) ) )
343, 32, 33sylanbrc 664 . . 3  |-  ( ph  ->  Fun  W )
35 funfn 5442 . . 3  |-  ( Fun 
W  <->  W  Fn  dom  W )
3634, 35sylib 196 . 2  |-  ( ph  ->  W  Fn  dom  W
)
37 vex 2970 . . . . 5  |-  s  e. 
_V
3837elrn 5075 . . . 4  |-  ( s  e.  ran  W  <->  E. w  w W s )
392releldmi 5071 . . . . . . . . . . . 12  |-  ( w W s  ->  w  e.  dom  W )
4039adantl 466 . . . . . . . . . . 11  |-  ( (
ph  /\  w W
s )  ->  w  e.  dom  W )
41 elssuni 4116 . . . . . . . . . . 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 3398 . . . . . . . . 9  |-  ( (
ph  /\  w W
s )  ->  w  C_  X )
45 xpss12 4940 . . . . . . . . 9  |-  ( ( w  C_  X  /\  w  C_  X )  -> 
( w  X.  w
)  C_  ( X  X.  X ) )
4644, 44, 45syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  w W
s )  ->  (
w  X.  w ) 
C_  ( X  X.  X ) )
4717, 46sstrd 3361 . . . . . . 7  |-  ( (
ph  /\  w W
s )  ->  s  C_  ( X  X.  X
) )
4847ex 434 . . . . . 6  |-  ( ph  ->  ( w W s  ->  s  C_  ( X  X.  X ) ) )
49 selpw 3862 . . . . . 6  |-  ( s  e.  ~P ( X  X.  X )  <->  s  C_  ( X  X.  X
) )
5048, 49syl6ibr 227 . . . . 5  |-  ( ph  ->  ( w W s  ->  s  e.  ~P ( X  X.  X
) ) )
5150exlimdv 1690 . . . 4  |-  ( ph  ->  ( E. w  w W s  ->  s  e.  ~P ( X  X.  X ) ) )
5238, 51syl5bi 217 . . 3  |-  ( ph  ->  ( s  e.  ran  W  ->  s  e.  ~P ( X  X.  X
) ) )
5352ssrdv 3357 . 2  |-  ( ph  ->  ran  W  C_  ~P ( X  X.  X
) )
54 df-f 5417 . 2  |-  ( W : dom  W --> ~P ( X  X.  X )  <->  ( W  Fn  dom  W  /\  ran  W 
C_  ~P ( X  X.  X ) ) )
5536, 53, 54sylanbrc 664 1  |-  ( ph  ->  W : dom  W --> ~P ( X  X.  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756   A.wral 2710   _Vcvv 2967   [.wsbc 3181    i^i cin 3322    C_ wss 3323   ~Pcpw 3855   {csn 3872   U.cuni 4086   class class class wbr 4287   {copab 4344    We wwe 4673    X. cxp 4833   `'ccnv 4834   dom cdm 4835   ran crn 4836   "cima 4838   Rel wrel 4840   Fun wfun 5407    Fn wfn 5408   -->wf 5409  (class class class)co 6086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-recs 6824  df-oi 7716
This theorem is referenced by:  fpwwe2lem13  8801  fpwwe2  8802
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