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Theorem pwfseqlem4 8431
Description: Lemma for pwfseq 8433. Derive a final contradiction from the function  F in pwfseqlem3 8429. Applying fpwwe2 8412 to it, we get a certain maximal well-ordered subset 
Z, but the defining property  ( Z F ( W `  Z
) )  e.  Z contradicts our assumption on  F, so we are reduced to the case of 
Z finite. This too is a contradiction, though, because  Z and its preimage under  ( W `  Z
) are distinct sets of the same cardinality and in a subset relation, which is impossible for finite sets. (Contributed by Mario Carneiro, 31-May-2015.)
Hypotheses
Ref Expression
pwfseqlem4.g  |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )
pwfseqlem4.x  |-  ( ph  ->  X  C_  A )
pwfseqlem4.h  |-  ( ph  ->  H : om -1-1-onto-> X )
pwfseqlem4.ps  |-  ( ps  <->  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  /\  om  ~<_  x ) )
pwfseqlem4.k  |-  ( (
ph  /\  ps )  ->  K : U_ n  e.  om  ( x  ^m  n ) -1-1-> x )
pwfseqlem4.d  |-  D  =  ( G `  {
w  e.  x  |  ( ( `' K `  w )  e.  ran  G  /\  -.  w  e.  ( `' G `  ( `' K `  w ) ) ) } )
pwfseqlem4.f  |-  F  =  ( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
pwfseqlem4.w  |-  W  =  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. b  e.  a 
[. ( `' s
" { b } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  b ) ) }
pwfseqlem4.z  |-  Z  = 
U. dom  W
Assertion
Ref Expression
pwfseqlem4  |-  -.  ph
Distinct variable groups:    n, r, w, x, z    D, n, z    a, b, s, v, F    w, G    w, K    r, a, x, z, H, b, s, v    n, a, ph, b, s, v, r, x, z    ps, n, z    A, a, n, r, s, x, z    W, a, b, s, v    Z, a, b, s, v
Allowed substitution hints:    ph( w)    ps( x, w, v, s, r, a, b)    A( w, v, b)    D( x, w, v, s, r, a, b)    F( x, z, w, n, r)    G( x, z, v, n, s, r, a, b)    H( w, n)    K( x, z, v, n, s, r, a, b)    W( x, z, w, n, r)    X( x, z, w, v, n, s, r, a, b)    Z( x, z, w, n, r)

Proof of Theorem pwfseqlem4
StepHypRef Expression
1 eqid 2366 . . . . . . . . . . 11  |-  Z  =  Z
2 eqid 2366 . . . . . . . . . . 11  |-  ( W `
 Z )  =  ( W `  Z
)
31, 2pm3.2i 441 . . . . . . . . . 10  |-  ( Z  =  Z  /\  ( W `  Z )  =  ( W `  Z ) )
4 pwfseqlem4.w . . . . . . . . . . 11  |-  W  =  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. b  e.  a 
[. ( `' s
" { b } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  b ) ) }
5 pwfseqlem4.g . . . . . . . . . . . . 13  |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )
6 omex 7491 . . . . . . . . . . . . . 14  |-  om  e.  _V
7 ovex 6006 . . . . . . . . . . . . . 14  |-  ( A  ^m  n )  e. 
_V
86, 7iunex 5891 . . . . . . . . . . . . 13  |-  U_ n  e.  om  ( A  ^m  n )  e.  _V
9 f1dmex 5871 . . . . . . . . . . . . 13  |-  ( ( G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n )  /\  U_ n  e.  om  ( A  ^m  n )  e. 
_V )  ->  ~P A  e.  _V )
105, 8, 9sylancl 643 . . . . . . . . . . . 12  |-  ( ph  ->  ~P A  e.  _V )
11 pwexb 4667 . . . . . . . . . . . 12  |-  ( A  e.  _V  <->  ~P A  e.  _V )
1210, 11sylibr 203 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  _V )
13 pwfseqlem4.x . . . . . . . . . . . 12  |-  ( ph  ->  X  C_  A )
14 pwfseqlem4.h . . . . . . . . . . . 12  |-  ( ph  ->  H : om -1-1-onto-> X )
15 pwfseqlem4.ps . . . . . . . . . . . 12  |-  ( ps  <->  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  /\  om  ~<_  x ) )
16 pwfseqlem4.k . . . . . . . . . . . 12  |-  ( (
ph  /\  ps )  ->  K : U_ n  e.  om  ( x  ^m  n ) -1-1-> x )
17 pwfseqlem4.d . . . . . . . . . . . 12  |-  D  =  ( G `  {
w  e.  x  |  ( ( `' K `  w )  e.  ran  G  /\  -.  w  e.  ( `' G `  ( `' K `  w ) ) ) } )
18 pwfseqlem4.f . . . . . . . . . . . 12  |-  F  =  ( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
195, 13, 14, 15, 16, 17, 18pwfseqlem4a 8430 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( a F s )  e.  A )
20 pwfseqlem4.z . . . . . . . . . . 11  |-  Z  = 
U. dom  W
214, 12, 19, 20fpwwe2 8412 . . . . . . . . . 10  |-  ( ph  ->  ( ( Z W ( W `  Z
)  /\  ( Z F ( W `  Z ) )  e.  Z )  <->  ( Z  =  Z  /\  ( W `  Z )  =  ( W `  Z ) ) ) )
223, 21mpbiri 224 . . . . . . . . 9  |-  ( ph  ->  ( Z W ( W `  Z )  /\  ( Z F ( W `  Z
) )  e.  Z
) )
2322simprd 449 . . . . . . . 8  |-  ( ph  ->  ( Z F ( W `  Z ) )  e.  Z )
2422simpld 445 . . . . . . . . . . . . 13  |-  ( ph  ->  Z W ( W `
 Z ) )
254, 12fpwwe2lem2 8401 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Z W ( W `  Z )  <-> 
( ( Z  C_  A  /\  ( W `  Z )  C_  ( Z  X.  Z ) )  /\  ( ( W `
 Z )  We  Z  /\  A. b  e.  Z  [. ( `' ( W `  Z
) " { b } )  /  v ]. ( v F ( ( W `  Z
)  i^i  ( v  X.  v ) ) )  =  b ) ) ) )
2624, 25mpbid 201 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( Z  C_  A  /\  ( W `  Z )  C_  ( Z  X.  Z ) )  /\  ( ( W `
 Z )  We  Z  /\  A. b  e.  Z  [. ( `' ( W `  Z
) " { b } )  /  v ]. ( v F ( ( W `  Z
)  i^i  ( v  X.  v ) ) )  =  b ) ) )
2726simpld 445 . . . . . . . . . . 11  |-  ( ph  ->  ( Z  C_  A  /\  ( W `  Z
)  C_  ( Z  X.  Z ) ) )
2827simpld 445 . . . . . . . . . 10  |-  ( ph  ->  Z  C_  A )
29 ssexg 4262 . . . . . . . . . 10  |-  ( ( Z  C_  A  /\  A  e.  _V )  ->  Z  e.  _V )
3028, 12, 29syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  Z  e.  _V )
31 sseq1 3285 . . . . . . . . . . . . . 14  |-  ( a  =  Z  ->  (
a  C_  A  <->  Z  C_  A
) )
32 id 19 . . . . . . . . . . . . . . . 16  |-  ( a  =  Z  ->  a  =  Z )
3332, 32xpeq12d 4817 . . . . . . . . . . . . . . 15  |-  ( a  =  Z  ->  (
a  X.  a )  =  ( Z  X.  Z ) )
3433sseq2d 3292 . . . . . . . . . . . . . 14  |-  ( a  =  Z  ->  (
( W `  Z
)  C_  ( a  X.  a )  <->  ( W `  Z )  C_  ( Z  X.  Z ) ) )
35 weeq2 4485 . . . . . . . . . . . . . 14  |-  ( a  =  Z  ->  (
( W `  Z
)  We  a  <->  ( W `  Z )  We  Z
) )
3631, 34, 353anbi123d 1253 . . . . . . . . . . . . 13  |-  ( a  =  Z  ->  (
( a  C_  A  /\  ( W `  Z
)  C_  ( a  X.  a )  /\  ( W `  Z )  We  a )  <->  ( Z  C_  A  /\  ( W `
 Z )  C_  ( Z  X.  Z
)  /\  ( W `  Z )  We  Z
) ) )
3736anbi2d 684 . . . . . . . . . . . 12  |-  ( a  =  Z  ->  (
( ph  /\  (
a  C_  A  /\  ( W `  Z ) 
C_  ( a  X.  a )  /\  ( W `  Z )  We  a ) )  <->  ( ph  /\  ( Z  C_  A  /\  ( W `  Z
)  C_  ( Z  X.  Z )  /\  ( W `  Z )  We  Z ) ) ) )
38 id 19 . . . . . . . . . . . . . . . 16  |-  ( ( Z  C_  A  /\  ( W `  Z ) 
C_  ( Z  X.  Z )  /\  ( W `  Z )  We  Z )  ->  ( Z  C_  A  /\  ( W `  Z )  C_  ( Z  X.  Z
)  /\  ( W `  Z )  We  Z
) )
39383expa 1152 . . . . . . . . . . . . . . 15  |-  ( ( ( Z  C_  A  /\  ( W `  Z
)  C_  ( Z  X.  Z ) )  /\  ( W `  Z )  We  Z )  -> 
( Z  C_  A  /\  ( W `  Z
)  C_  ( Z  X.  Z )  /\  ( W `  Z )  We  Z ) )
4039adantrr 697 . . . . . . . . . . . . . 14  |-  ( ( ( Z  C_  A  /\  ( W `  Z
)  C_  ( Z  X.  Z ) )  /\  ( ( W `  Z )  We  Z  /\  A. b  e.  Z  [. ( `' ( W `
 Z ) " { b } )  /  v ]. (
v F ( ( W `  Z )  i^i  ( v  X.  v ) ) )  =  b ) )  ->  ( Z  C_  A  /\  ( W `  Z )  C_  ( Z  X.  Z )  /\  ( W `  Z )  We  Z ) )
4126, 40syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Z  C_  A  /\  ( W `  Z
)  C_  ( Z  X.  Z )  /\  ( W `  Z )  We  Z ) )
4241pm4.71i 613 . . . . . . . . . . . 12  |-  ( ph  <->  (
ph  /\  ( Z  C_  A  /\  ( W `
 Z )  C_  ( Z  X.  Z
)  /\  ( W `  Z )  We  Z
) ) )
4337, 42syl6bbr 254 . . . . . . . . . . 11  |-  ( a  =  Z  ->  (
( ph  /\  (
a  C_  A  /\  ( W `  Z ) 
C_  ( a  X.  a )  /\  ( W `  Z )  We  a ) )  <->  ph ) )
44 oveq1 5988 . . . . . . . . . . . . 13  |-  ( a  =  Z  ->  (
a F ( W `
 Z ) )  =  ( Z F ( W `  Z
) ) )
4544, 32eleq12d 2434 . . . . . . . . . . . 12  |-  ( a  =  Z  ->  (
( a F ( W `  Z ) )  e.  a  <->  ( Z F ( W `  Z ) )  e.  Z ) )
46 breq1 4128 . . . . . . . . . . . 12  |-  ( a  =  Z  ->  (
a  ~<  om  <->  Z  ~<  om )
)
4745, 46imbi12d 311 . . . . . . . . . . 11  |-  ( a  =  Z  ->  (
( ( a F ( W `  Z
) )  e.  a  ->  a  ~<  om )  <->  ( ( Z F ( W `  Z ) )  e.  Z  ->  Z  ~<  om ) ) )
4843, 47imbi12d 311 . . . . . . . . . 10  |-  ( a  =  Z  ->  (
( ( ph  /\  ( a  C_  A  /\  ( W `  Z
)  C_  ( a  X.  a )  /\  ( W `  Z )  We  a ) )  -> 
( ( a F ( W `  Z
) )  e.  a  ->  a  ~<  om )
)  <->  ( ph  ->  ( ( Z F ( W `  Z ) )  e.  Z  ->  Z  ~<  om ) ) ) )
49 fvex 5646 . . . . . . . . . . 11  |-  ( W `
 Z )  e. 
_V
50 sseq1 3285 . . . . . . . . . . . . . 14  |-  ( s  =  ( W `  Z )  ->  (
s  C_  ( a  X.  a )  <->  ( W `  Z )  C_  (
a  X.  a ) ) )
51 weeq1 4484 . . . . . . . . . . . . . 14  |-  ( s  =  ( W `  Z )  ->  (
s  We  a  <->  ( W `  Z )  We  a
) )
5250, 513anbi23d 1256 . . . . . . . . . . . . 13  |-  ( s  =  ( W `  Z )  ->  (
( a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  <->  ( a  C_  A  /\  ( W `  Z ) 
C_  ( a  X.  a )  /\  ( W `  Z )  We  a ) ) )
5352anbi2d 684 . . . . . . . . . . . 12  |-  ( s  =  ( W `  Z )  ->  (
( ph  /\  (
a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a ) )  <->  ( ph  /\  ( a  C_  A  /\  ( W `  Z
)  C_  ( a  X.  a )  /\  ( W `  Z )  We  a ) ) ) )
54 oveq2 5989 . . . . . . . . . . . . . 14  |-  ( s  =  ( W `  Z )  ->  (
a F s )  =  ( a F ( W `  Z
) ) )
5554eleq1d 2432 . . . . . . . . . . . . 13  |-  ( s  =  ( W `  Z )  ->  (
( a F s )  e.  a  <->  ( a F ( W `  Z ) )  e.  a ) )
5655imbi1d 308 . . . . . . . . . . . 12  |-  ( s  =  ( W `  Z )  ->  (
( ( a F s )  e.  a  ->  a  ~<  om )  <->  ( ( a F ( W `  Z ) )  e.  a  -> 
a  ~<  om ) ) )
5753, 56imbi12d 311 . . . . . . . . . . 11  |-  ( s  =  ( W `  Z )  ->  (
( ( ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )
)  ->  ( (
a F s )  e.  a  ->  a  ~<  om ) )  <->  ( ( ph  /\  ( a  C_  A  /\  ( W `  Z )  C_  (
a  X.  a )  /\  ( W `  Z )  We  a
) )  ->  (
( a F ( W `  Z ) )  e.  a  -> 
a  ~<  om ) ) ) )
58 omelon 7494 . . . . . . . . . . . . . . 15  |-  om  e.  On
59 onenon 7729 . . . . . . . . . . . . . . 15  |-  ( om  e.  On  ->  om  e.  dom  card )
6058, 59ax-mp 8 . . . . . . . . . . . . . 14  |-  om  e.  dom  card
61 simpr3 964 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
s  We  a )
62 19.8a 1752 . . . . . . . . . . . . . . . 16  |-  ( s  We  a  ->  E. s 
s  We  a )
6361, 62syl 15 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  ->  E. s  s  We  a )
64 ween 7809 . . . . . . . . . . . . . . 15  |-  ( a  e.  dom  card  <->  E. s 
s  We  a )
6563, 64sylibr 203 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
a  e.  dom  card )
66 domtri2 7769 . . . . . . . . . . . . . 14  |-  ( ( om  e.  dom  card  /\  a  e.  dom  card )  ->  ( om  ~<_  a  <->  -.  a  ~<  om ) )
6760, 65, 66sylancr 644 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( om  ~<_  a  <->  -.  a  ~<  om ) )
68 nfv 1624 . . . . . . . . . . . . . . . . 17  |-  F/ r ( ph  /\  (
( a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  /\  om  ~<_  a ) )
69 nfcv 2502 . . . . . . . . . . . . . . . . . . 19  |-  F/_ r
a
70 nfmpt22 6041 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ r
( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
7118, 70nfcxfr 2499 . . . . . . . . . . . . . . . . . . 19  |-  F/_ r F
72 nfcv 2502 . . . . . . . . . . . . . . . . . . 19  |-  F/_ r
s
7369, 71, 72nfov 6004 . . . . . . . . . . . . . . . . . 18  |-  F/_ r
( a F s )
7473nfel1 2512 . . . . . . . . . . . . . . . . 17  |-  F/ r ( a F s )  e.  ( A 
\  a )
7568, 74nfim 1820 . . . . . . . . . . . . . . . 16  |-  F/ r ( ( ph  /\  ( ( a  C_  A  /\  s  C_  (
a  X.  a )  /\  s  We  a
)  /\  om  ~<_  a ) )  ->  ( a F s )  e.  ( A  \  a
) )
76 sseq1 3285 . . . . . . . . . . . . . . . . . . . 20  |-  ( r  =  s  ->  (
r  C_  ( a  X.  a )  <->  s  C_  ( a  X.  a
) ) )
77 weeq1 4484 . . . . . . . . . . . . . . . . . . . 20  |-  ( r  =  s  ->  (
r  We  a  <->  s  We  a ) )
7876, 773anbi23d 1256 . . . . . . . . . . . . . . . . . . 19  |-  ( r  =  s  ->  (
( a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  <->  ( a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a ) ) )
7978anbi1d 685 . . . . . . . . . . . . . . . . . 18  |-  ( r  =  s  ->  (
( ( a  C_  A  /\  r  C_  (
a  X.  a )  /\  r  We  a
)  /\  om  ~<_  a )  <-> 
( ( a  C_  A  /\  s  C_  (
a  X.  a )  /\  s  We  a
)  /\  om  ~<_  a ) ) )
8079anbi2d 684 . . . . . . . . . . . . . . . . 17  |-  ( r  =  s  ->  (
( ph  /\  (
( a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  /\  om  ~<_  a ) )  <-> 
( ph  /\  (
( a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  /\  om  ~<_  a ) ) ) )
81 oveq2 5989 . . . . . . . . . . . . . . . . . 18  |-  ( r  =  s  ->  (
a F r )  =  ( a F s ) )
8281eleq1d 2432 . . . . . . . . . . . . . . . . 17  |-  ( r  =  s  ->  (
( a F r )  e.  ( A 
\  a )  <->  ( a F s )  e.  ( A  \  a
) ) )
8380, 82imbi12d 311 . . . . . . . . . . . . . . . 16  |-  ( r  =  s  ->  (
( ( ph  /\  ( ( a  C_  A  /\  r  C_  (
a  X.  a )  /\  r  We  a
)  /\  om  ~<_  a ) )  ->  ( a F r )  e.  ( A  \  a
) )  <->  ( ( ph  /\  ( ( a 
C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a )  /\  om  ~<_  a ) )  -> 
( a F s )  e.  ( A 
\  a ) ) ) )
84 nfv 1624 . . . . . . . . . . . . . . . . . 18  |-  F/ x
( ph  /\  (
( a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  /\  om  ~<_  a ) )
85 nfcv 2502 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ x
a
86 nfmpt21 6040 . . . . . . . . . . . . . . . . . . . . 21  |-  F/_ x
( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
8718, 86nfcxfr 2499 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ x F
88 nfcv 2502 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ x
r
8985, 87, 88nfov 6004 . . . . . . . . . . . . . . . . . . 19  |-  F/_ x
( a F r )
9089nfel1 2512 . . . . . . . . . . . . . . . . . 18  |-  F/ x
( a F r )  e.  ( A 
\  a )
9184, 90nfim 1820 . . . . . . . . . . . . . . . . 17  |-  F/ x
( ( ph  /\  ( ( a  C_  A  /\  r  C_  (
a  X.  a )  /\  r  We  a
)  /\  om  ~<_  a ) )  ->  ( a F r )  e.  ( A  \  a
) )
92 sseq1 3285 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  =  a  ->  (
x  C_  A  <->  a  C_  A ) )
93 xpeq12 4811 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( x  =  a  /\  x  =  a )  ->  ( x  X.  x
)  =  ( a  X.  a ) )
9493anidms 626 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  =  a  ->  (
x  X.  x )  =  ( a  X.  a ) )
9594sseq2d 3292 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  =  a  ->  (
r  C_  ( x  X.  x )  <->  r  C_  ( a  X.  a
) ) )
96 weeq2 4485 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  =  a  ->  (
r  We  x  <->  r  We  a ) )
9792, 95, 963anbi123d 1253 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  =  a  ->  (
( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  <->  ( a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a ) ) )
98 breq2 4129 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  =  a  ->  ( om 
~<_  x  <->  om  ~<_  a ) )
9997, 98anbi12d 691 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  a  ->  (
( ( x  C_  A  /\  r  C_  (
x  X.  x )  /\  r  We  x
)  /\  om  ~<_  x )  <-> 
( ( a  C_  A  /\  r  C_  (
a  X.  a )  /\  r  We  a
)  /\  om  ~<_  a ) ) )
10015, 99syl5bb 248 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  a  ->  ( ps 
<->  ( ( a  C_  A  /\  r  C_  (
a  X.  a )  /\  r  We  a
)  /\  om  ~<_  a ) ) )
101100anbi2d 684 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  a  ->  (
( ph  /\  ps )  <->  (
ph  /\  ( (
a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  /\  om  ~<_  a ) ) ) )
102 oveq1 5988 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  a  ->  (
x F r )  =  ( a F r ) )
103 difeq2 3375 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  a  ->  ( A  \  x )  =  ( A  \  a
) )
104102, 103eleq12d 2434 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  a  ->  (
( x F r )  e.  ( A 
\  x )  <->  ( a F r )  e.  ( A  \  a
) ) )
105101, 104imbi12d 311 . . . . . . . . . . . . . . . . 17  |-  ( x  =  a  ->  (
( ( ph  /\  ps )  ->  ( x F r )  e.  ( A  \  x
) )  <->  ( ( ph  /\  ( ( a 
C_  A  /\  r  C_  ( a  X.  a
)  /\  r  We  a )  /\  om  ~<_  a ) )  -> 
( a F r )  e.  ( A 
\  a ) ) ) )
1065, 13, 14, 15, 16, 17, 18pwfseqlem3 8429 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ps )  ->  ( x F r )  e.  ( A 
\  x ) )
10791, 105, 106chvar 1999 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  /\  om  ~<_  a ) )  -> 
( a F r )  e.  ( A 
\  a ) )
10875, 83, 107chvar 1999 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  /\  om  ~<_  a ) )  -> 
( a F s )  e.  ( A 
\  a ) )
109 eldifn 3386 . . . . . . . . . . . . . . 15  |-  ( ( a F s )  e.  ( A  \ 
a )  ->  -.  ( a F s )  e.  a )
110108, 109syl 15 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( (
a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  /\  om  ~<_  a ) )  ->  -.  ( a F s )  e.  a )
111110expr 598 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( om  ~<_  a  ->  -.  ( a F s )  e.  a ) )
11267, 111sylbird 226 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( -.  a  ~<  om  ->  -.  ( a F s )  e.  a ) )
113112con4d 97 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( ( a F s )  e.  a  ->  a  ~<  om )
)
11449, 57, 113vtocl 2923 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  C_  A  /\  ( W `
 Z )  C_  ( a  X.  a
)  /\  ( W `  Z )  We  a
) )  ->  (
( a F ( W `  Z ) )  e.  a  -> 
a  ~<  om ) )
11548, 114vtoclg 2928 . . . . . . . . 9  |-  ( Z  e.  _V  ->  ( ph  ->  ( ( Z F ( W `  Z ) )  e.  Z  ->  Z  ~<  om ) ) )
11630, 115mpcom 32 . . . . . . . 8  |-  ( ph  ->  ( ( Z F ( W `  Z
) )  e.  Z  ->  Z  ~<  om )
)
11723, 116mpd 14 . . . . . . 7  |-  ( ph  ->  Z  ~<  om )
118 isfinite 7500 . . . . . . 7  |-  ( Z  e.  Fin  <->  Z  ~<  om )
119117, 118sylibr 203 . . . . . 6  |-  ( ph  ->  Z  e.  Fin )
1205, 13, 14, 15, 16, 17, 18pwfseqlem2 8428 . . . . . 6  |-  ( ( Z  e.  Fin  /\  ( W `  Z )  e.  _V )  -> 
( Z F ( W `  Z ) )  =  ( H `
 ( card `  Z
) ) )
121119, 49, 120sylancl 643 . . . . 5  |-  ( ph  ->  ( Z F ( W `  Z ) )  =  ( H `
 ( card `  Z
) ) )
122121, 23eqeltrrd 2441 . . . 4  |-  ( ph  ->  ( H `  ( card `  Z ) )  e.  Z )
1234, 12, 24fpwwe2lem3 8402 . . . . . . . . . 10  |-  ( (
ph  /\  ( H `  ( card `  Z
) )  e.  Z
)  ->  ( ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } ) F ( ( W `  Z
)  i^i  ( ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  X.  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } ) ) ) )  =  ( H `
 ( card `  Z
) ) )
124122, 123mpdan 649 . . . . . . . . 9  |-  ( ph  ->  ( ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) F ( ( W `
 Z )  i^i  ( ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } )  X.  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) ) ) )  =  ( H `  ( card `  Z ) ) )
125 cnvimass 5136 . . . . . . . . . . . 12  |-  ( `' ( W `  Z
) " { ( H `  ( card `  Z ) ) } )  C_  dom  ( W `
 Z )
12627simprd 449 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( W `  Z
)  C_  ( Z  X.  Z ) )
127 dmss 4981 . . . . . . . . . . . . . 14  |-  ( ( W `  Z ) 
C_  ( Z  X.  Z )  ->  dom  ( W `  Z ) 
C_  dom  ( Z  X.  Z ) )
128126, 127syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  ( W `  Z )  C_  dom  ( Z  X.  Z
) )
129 dmxpss 5210 . . . . . . . . . . . . 13  |-  dom  ( Z  X.  Z )  C_  Z
130128, 129syl6ss 3277 . . . . . . . . . . . 12  |-  ( ph  ->  dom  ( W `  Z )  C_  Z
)
131125, 130syl5ss 3276 . . . . . . . . . . 11  |-  ( ph  ->  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  C_  Z )
132 ssfi 7226 . . . . . . . . . . 11  |-  ( ( Z  e.  Fin  /\  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  C_  Z
)  ->  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } )  e.  Fin )
133119, 131, 132syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  e. 
Fin )
13449inex1 4257 . . . . . . . . . 10  |-  ( ( W `  Z )  i^i  ( ( `' ( W `  Z
) " { ( H `  ( card `  Z ) ) } )  X.  ( `' ( W `  Z
) " { ( H `  ( card `  Z ) ) } ) ) )  e. 
_V
1355, 13, 14, 15, 16, 17, 18pwfseqlem2 8428 . . . . . . . . . 10  |-  ( ( ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  e. 
Fin  /\  ( ( W `  Z )  i^i  ( ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } )  X.  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) ) )  e.  _V )  ->  ( ( `' ( W `  Z
) " { ( H `  ( card `  Z ) ) } ) F ( ( W `  Z )  i^i  ( ( `' ( W `  Z
) " { ( H `  ( card `  Z ) ) } )  X.  ( `' ( W `  Z
) " { ( H `  ( card `  Z ) ) } ) ) ) )  =  ( H `  ( card `  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) ) ) )
136133, 134, 135sylancl 643 . . . . . . . . 9  |-  ( ph  ->  ( ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) F ( ( W `
 Z )  i^i  ( ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } )  X.  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) ) ) )  =  ( H `  ( card `  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) ) ) )
137124, 136eqtr3d 2400 . . . . . . . 8  |-  ( ph  ->  ( H `  ( card `  Z ) )  =  ( H `  ( card `  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) ) ) )
138 f1of1 5577 . . . . . . . . . 10  |-  ( H : om -1-1-onto-> X  ->  H : om
-1-1-> X )
13914, 138syl 15 . . . . . . . . 9  |-  ( ph  ->  H : om -1-1-> X
)
140 ficardom 7741 . . . . . . . . . 10  |-  ( Z  e.  Fin  ->  ( card `  Z )  e. 
om )
141119, 140syl 15 . . . . . . . . 9  |-  ( ph  ->  ( card `  Z
)  e.  om )
142 ficardom 7741 . . . . . . . . . 10  |-  ( ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  e.  Fin  ->  ( card `  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } ) )  e. 
om )
143133, 142syl 15 . . . . . . . . 9  |-  ( ph  ->  ( card `  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } ) )  e. 
om )
144 f1fveq 5908 . . . . . . . . 9  |-  ( ( H : om -1-1-> X  /\  ( ( card `  Z
)  e.  om  /\  ( card `  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) )  e.  om )
)  ->  ( ( H `  ( card `  Z ) )  =  ( H `  ( card `  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) ) )  <->  ( card `  Z )  =  (
card `  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) ) ) )
145139, 141, 143, 144syl12anc 1181 . . . . . . . 8  |-  ( ph  ->  ( ( H `  ( card `  Z )
)  =  ( H `
 ( card `  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } ) ) )  <-> 
( card `  Z )  =  ( card `  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } ) ) ) )
146137, 145mpbid 201 . . . . . . 7  |-  ( ph  ->  ( card `  Z
)  =  ( card `  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } ) ) )
147146eqcomd 2371 . . . . . 6  |-  ( ph  ->  ( card `  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } ) )  =  ( card `  Z
) )
148 finnum 7728 . . . . . . . 8  |-  ( ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  e.  Fin  ->  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  e. 
dom  card )
149133, 148syl 15 . . . . . . 7  |-  ( ph  ->  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  e. 
dom  card )
150 finnum 7728 . . . . . . . 8  |-  ( Z  e.  Fin  ->  Z  e.  dom  card )
151119, 150syl 15 . . . . . . 7  |-  ( ph  ->  Z  e.  dom  card )
152 carden2 7767 . . . . . . 7  |-  ( ( ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  e. 
dom  card  /\  Z  e.  dom  card )  ->  (
( card `  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) )  =  ( card `  Z )  <->  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) 
~~  Z ) )
153149, 151, 152syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( card `  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } ) )  =  ( card `  Z
)  <->  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) 
~~  Z ) )
154147, 153mpbid 201 . . . . 5  |-  ( ph  ->  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  ~~  Z )
155 dfpss2 3348 . . . . . . . 8  |-  ( ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  C.  Z  <->  ( ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  C_  Z  /\  -.  ( `' ( W `  Z
) " { ( H `  ( card `  Z ) ) } )  =  Z ) )
156155baib 871 . . . . . . 7  |-  ( ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  C_  Z  ->  ( ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) 
C.  Z  <->  -.  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  =  Z ) )
157131, 156syl 15 . . . . . 6  |-  ( ph  ->  ( ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) 
C.  Z  <->  -.  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  =  Z ) )
158 php3 7190 . . . . . . . . 9  |-  ( ( Z  e.  Fin  /\  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  C.  Z
)  ->  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) 
~<  Z )
159 sdomnen 7033 . . . . . . . . 9  |-  ( ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  ~<  Z  ->  -.  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  ~~  Z )
160158, 159syl 15 . . . . . . . 8  |-  ( ( Z  e.  Fin  /\  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  C.  Z
)  ->  -.  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  ~~  Z
)
161160ex 423 . . . . . . 7  |-  ( Z  e.  Fin  ->  (
( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  C.  Z  ->  -.  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) 
~~  Z ) )
162119, 161syl 15 . . . . . 6  |-  ( ph  ->  ( ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) 
C.  Z  ->  -.  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  ~~  Z
) )
163157, 162sylbird 226 . . . . 5  |-  ( ph  ->  ( -.  ( `' ( W `  Z
) " { ( H `  ( card `  Z ) ) } )  =  Z  ->  -.  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  ~~  Z ) )
164154, 163mt4d 130 . . . 4  |-  ( ph  ->  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  =  Z )
165122, 164eleqtrrd 2443 . . 3  |-  ( ph  ->  ( H `  ( card `  Z ) )  e.  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) )
166 fvex 5646 . . . 4  |-  ( H `
 ( card `  Z
) )  e.  _V
167166eliniseg 5145 . . . 4  |-  ( ( H `  ( card `  Z ) )  e. 
_V  ->  ( ( H `
 ( card `  Z
) )  e.  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  <->  ( H `  ( card `  Z
) ) ( W `
 Z ) ( H `  ( card `  Z ) ) ) )
168166, 167ax-mp 8 . . 3  |-  ( ( H `  ( card `  Z ) )  e.  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  <->  ( H `  ( card `  Z
) ) ( W `
 Z ) ( H `  ( card `  Z ) ) )
169165, 168sylib 188 . 2  |-  ( ph  ->  ( H `  ( card `  Z ) ) ( W `  Z
) ( H `  ( card `  Z )
) )
17026simprd 449 . . . . 5  |-  ( ph  ->  ( ( W `  Z )  We  Z  /\  A. b  e.  Z  [. ( `' ( W `
 Z ) " { b } )  /  v ]. (
v F ( ( W `  Z )  i^i  ( v  X.  v ) ) )  =  b ) )
171170simpld 445 . . . 4  |-  ( ph  ->  ( W `  Z
)  We  Z )
172 weso 4487 . . . 4  |-  ( ( W `  Z )  We  Z  ->  ( W `  Z )  Or  Z )
173171, 172syl 15 . . 3  |-  ( ph  ->  ( W `  Z
)  Or  Z )
174 sonr 4438 . . 3  |-  ( ( ( W `  Z
)  Or  Z  /\  ( H `  ( card `  Z ) )  e.  Z )  ->  -.  ( H `  ( card `  Z ) ) ( W `  Z ) ( H `  ( card `  Z ) ) )
175173, 122, 174syl2anc 642 . 2  |-  ( ph  ->  -.  ( H `  ( card `  Z )
) ( W `  Z ) ( H `
 ( card `  Z
) ) )
176169, 175pm2.65i 165 1  |-  -.  ph
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935   E.wex 1546    = wceq 1647    e. wcel 1715   A.wral 2628   {crab 2632   _Vcvv 2873   [.wsbc 3077    \ cdif 3235    i^i cin 3237    C_ wss 3238    C. wpss 3239   ifcif 3654   ~Pcpw 3714   {csn 3729   U.cuni 3929   |^|cint 3964   U_ciun 4007   class class class wbr 4125   {copab 4178    Or wor 4416    We wwe 4454   Oncon0 4495   omcom 4759    X. cxp 4790   `'ccnv 4791   dom cdm 4792   ran crn 4793   "cima 4795   -1-1->wf1 5355   -1-1-onto->wf1o 5357   ` cfv 5358  (class class class)co 5981    e. cmpt2 5983    ^m cmap 6915    ~~ cen 7003    ~<_ cdom 7004    ~< csdm 7005   Fincfn 7006   cardccrd 7715
This theorem is referenced by:  pwfseqlem5  8432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-oi 7372  df-card 7719
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