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Theorem pwfseqlem4a 9051
Description: Lemma for pwfseqlem4 9052. (Contributed by Mario Carneiro, 7-Jun-2016.)
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 } ) ) )
Assertion
Ref Expression
pwfseqlem4a  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( a F s )  e.  A )
Distinct variable groups:    n, r, w, x, z    D, n, z    s, a, F   
w, G    w, K    r, a, x, z, H, s    n, a, ph, s, r, x, z    ps, n, z    A, a, n, r, s, x, z
Allowed substitution hints:    ph( w)    ps( x, w, s, r, a)    A( w)    D( x, w, s, r, a)    F( x, z, w, n, r)    G( x, z, n, s, r, a)    H( w, n)    K( x, z, n, s, r, a)    X( x, z, w, n, s, r, a)

Proof of Theorem pwfseqlem4a
StepHypRef Expression
1 isfinite 8081 . . 3  |-  ( a  e.  Fin  <->  a  ~<  om )
2 simpr 461 . . . . . . 7  |-  ( (
ph  /\  a  e.  Fin )  ->  a  e. 
Fin )
3 vex 3121 . . . . . . 7  |-  s  e. 
_V
4 pwfseqlem4.g . . . . . . . 8  |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )
5 pwfseqlem4.x . . . . . . . 8  |-  ( ph  ->  X  C_  A )
6 pwfseqlem4.h . . . . . . . 8  |-  ( ph  ->  H : om -1-1-onto-> X )
7 pwfseqlem4.ps . . . . . . . 8  |-  ( ps  <->  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  /\  om  ~<_  x ) )
8 pwfseqlem4.k . . . . . . . 8  |-  ( (
ph  /\  ps )  ->  K : U_ n  e.  om  ( x  ^m  n ) -1-1-> x )
9 pwfseqlem4.d . . . . . . . 8  |-  D  =  ( G `  {
w  e.  x  |  ( ( `' K `  w )  e.  ran  G  /\  -.  w  e.  ( `' G `  ( `' K `  w ) ) ) } )
10 pwfseqlem4.f . . . . . . . 8  |-  F  =  ( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
114, 5, 6, 7, 8, 9, 10pwfseqlem2 9049 . . . . . . 7  |-  ( ( a  e.  Fin  /\  s  e.  _V )  ->  ( a F s )  =  ( H `
 ( card `  a
) ) )
122, 3, 11sylancl 662 . . . . . 6  |-  ( (
ph  /\  a  e.  Fin )  ->  ( a F s )  =  ( H `  ( card `  a ) ) )
13 f1of 5822 . . . . . . . . 9  |-  ( H : om -1-1-onto-> X  ->  H : om
--> X )
146, 13syl 16 . . . . . . . 8  |-  ( ph  ->  H : om --> X )
15 fss 5745 . . . . . . . 8  |-  ( ( H : om --> X  /\  X  C_  A )  ->  H : om --> A )
1614, 5, 15syl2anc 661 . . . . . . 7  |-  ( ph  ->  H : om --> A )
17 ficardom 8354 . . . . . . 7  |-  ( a  e.  Fin  ->  ( card `  a )  e. 
om )
18 ffvelrn 6030 . . . . . . 7  |-  ( ( H : om --> A  /\  ( card `  a )  e.  om )  ->  ( H `  ( card `  a ) )  e.  A )
1916, 17, 18syl2an 477 . . . . . 6  |-  ( (
ph  /\  a  e.  Fin )  ->  ( H `
 ( card `  a
) )  e.  A
)
2012, 19eqeltrd 2555 . . . . 5  |-  ( (
ph  /\  a  e.  Fin )  ->  ( a F s )  e.  A )
2120ex 434 . . . 4  |-  ( ph  ->  ( a  e.  Fin  ->  ( a F s )  e.  A ) )
2221adantr 465 . . 3  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( a  e.  Fin  ->  ( a F s )  e.  A ) )
231, 22syl5bir 218 . 2  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( a  ~<  om  ->  ( a F s )  e.  A ) )
24 omelon 8075 . . . . 5  |-  om  e.  On
25 onenon 8342 . . . . 5  |-  ( om  e.  On  ->  om  e.  dom  card )
2624, 25ax-mp 5 . . . 4  |-  om  e.  dom  card
27 simpr3 1004 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
s  We  a )
28 19.8a 1806 . . . . . 6  |-  ( s  We  a  ->  E. s 
s  We  a )
2927, 28syl 16 . . . . 5  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  ->  E. s  s  We  a )
30 ween 8428 . . . . 5  |-  ( a  e.  dom  card  <->  E. s 
s  We  a )
3129, 30sylibr 212 . . . 4  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
a  e.  dom  card )
32 domtri2 8382 . . . 4  |-  ( ( om  e.  dom  card  /\  a  e.  dom  card )  ->  ( om  ~<_  a  <->  -.  a  ~<  om ) )
3326, 31, 32sylancr 663 . . 3  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( om  ~<_  a  <->  -.  a  ~<  om ) )
34 nfv 1683 . . . . . . 7  |-  F/ r ( ph  /\  (
( a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  /\  om  ~<_  a ) )
35 nfcv 2629 . . . . . . . . 9  |-  F/_ r
a
36 nfmpt22 6360 . . . . . . . . . 10  |-  F/_ r
( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
3710, 36nfcxfr 2627 . . . . . . . . 9  |-  F/_ r F
38 nfcv 2629 . . . . . . . . 9  |-  F/_ r
s
3935, 37, 38nfov 6318 . . . . . . . 8  |-  F/_ r
( a F s )
4039nfel1 2645 . . . . . . 7  |-  F/ r ( a F s )  e.  ( A 
\  a )
4134, 40nfim 1867 . . . . . 6  |-  F/ r ( ( ph  /\  ( ( a  C_  A  /\  s  C_  (
a  X.  a )  /\  s  We  a
)  /\  om  ~<_  a ) )  ->  ( a F s )  e.  ( A  \  a
) )
42 sseq1 3530 . . . . . . . . . 10  |-  ( r  =  s  ->  (
r  C_  ( a  X.  a )  <->  s  C_  ( a  X.  a
) ) )
43 weeq1 4873 . . . . . . . . . 10  |-  ( r  =  s  ->  (
r  We  a  <->  s  We  a ) )
4442, 433anbi23d 1302 . . . . . . . . 9  |-  ( r  =  s  ->  (
( a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  <->  ( a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a ) ) )
4544anbi1d 704 . . . . . . . 8  |-  ( 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 ) ) )
4645anbi2d 703 . . . . . . 7  |-  ( 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 ) ) ) )
47 oveq2 6303 . . . . . . . 8  |-  ( r  =  s  ->  (
a F r )  =  ( a F s ) )
4847eleq1d 2536 . . . . . . 7  |-  ( r  =  s  ->  (
( a F r )  e.  ( A 
\  a )  <->  ( a F s )  e.  ( A  \  a
) ) )
4946, 48imbi12d 320 . . . . . 6  |-  ( 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 ) ) ) )
50 nfv 1683 . . . . . . . 8  |-  F/ x
( ph  /\  (
( a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  /\  om  ~<_  a ) )
51 nfcv 2629 . . . . . . . . . 10  |-  F/_ x
a
52 nfmpt21 6359 . . . . . . . . . . 11  |-  F/_ x
( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
5310, 52nfcxfr 2627 . . . . . . . . . 10  |-  F/_ x F
54 nfcv 2629 . . . . . . . . . 10  |-  F/_ x
r
5551, 53, 54nfov 6318 . . . . . . . . 9  |-  F/_ x
( a F r )
5655nfel1 2645 . . . . . . . 8  |-  F/ x
( a F r )  e.  ( A 
\  a )
5750, 56nfim 1867 . . . . . . 7  |-  F/ x
( ( ph  /\  ( ( a  C_  A  /\  r  C_  (
a  X.  a )  /\  r  We  a
)  /\  om  ~<_  a ) )  ->  ( a F r )  e.  ( A  \  a
) )
58 sseq1 3530 . . . . . . . . . . . 12  |-  ( x  =  a  ->  (
x  C_  A  <->  a  C_  A ) )
59 xpeq12 5024 . . . . . . . . . . . . . 14  |-  ( ( x  =  a  /\  x  =  a )  ->  ( x  X.  x
)  =  ( a  X.  a ) )
6059anidms 645 . . . . . . . . . . . . 13  |-  ( x  =  a  ->  (
x  X.  x )  =  ( a  X.  a ) )
6160sseq2d 3537 . . . . . . . . . . . 12  |-  ( x  =  a  ->  (
r  C_  ( x  X.  x )  <->  r  C_  ( a  X.  a
) ) )
62 weeq2 4874 . . . . . . . . . . . 12  |-  ( x  =  a  ->  (
r  We  x  <->  r  We  a ) )
6358, 61, 623anbi123d 1299 . . . . . . . . . . 11  |-  ( x  =  a  ->  (
( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  <->  ( a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a ) ) )
64 breq2 4457 . . . . . . . . . . 11  |-  ( x  =  a  ->  ( om 
~<_  x  <->  om  ~<_  a ) )
6563, 64anbi12d 710 . . . . . . . . . 10  |-  ( 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 ) ) )
667, 65syl5bb 257 . . . . . . . . 9  |-  ( x  =  a  ->  ( ps 
<->  ( ( a  C_  A  /\  r  C_  (
a  X.  a )  /\  r  We  a
)  /\  om  ~<_  a ) ) )
6766anbi2d 703 . . . . . . . 8  |-  ( x  =  a  ->  (
( ph  /\  ps )  <->  (
ph  /\  ( (
a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  /\  om  ~<_  a ) ) ) )
68 oveq1 6302 . . . . . . . . 9  |-  ( x  =  a  ->  (
x F r )  =  ( a F r ) )
69 difeq2 3621 . . . . . . . . 9  |-  ( x  =  a  ->  ( A  \  x )  =  ( A  \  a
) )
7068, 69eleq12d 2549 . . . . . . . 8  |-  ( x  =  a  ->  (
( x F r )  e.  ( A 
\  x )  <->  ( a F r )  e.  ( A  \  a
) ) )
7167, 70imbi12d 320 . . . . . . 7  |-  ( 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 ) ) ) )
724, 5, 6, 7, 8, 9, 10pwfseqlem3 9050 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  ( x F r )  e.  ( A 
\  x ) )
7357, 71, 72chvar 1982 . . . . . 6  |-  ( (
ph  /\  ( (
a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  /\  om  ~<_  a ) )  -> 
( a F r )  e.  ( A 
\  a ) )
7441, 49, 73chvar 1982 . . . . 5  |-  ( (
ph  /\  ( (
a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  /\  om  ~<_  a ) )  -> 
( a F s )  e.  ( A 
\  a ) )
7574eldifad 3493 . . . 4  |-  ( (
ph  /\  ( (
a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  /\  om  ~<_  a ) )  -> 
( a F s )  e.  A )
7675expr 615 . . 3  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( om  ~<_  a  -> 
( a F s )  e.  A ) )
7733, 76sylbird 235 . 2  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( -.  a  ~<  om  ->  ( a F s )  e.  A
) )
7823, 77pm2.61d 158 1  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( a F s )  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   {crab 2821   _Vcvv 3118    \ cdif 3478    C_ wss 3481   ifcif 3945   ~Pcpw 4016   |^|cint 4288   U_ciun 4331   class class class wbr 4453    We wwe 4843   Oncon0 4884    X. cxp 5003   `'ccnv 5004   dom cdm 5005   ran crn 5006   -->wf 5590   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   omcom 6695    ^m cmap 7432    ~<_ cdom 7526    ~< csdm 7527   Fincfn 7528   cardccrd 8328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332
This theorem is referenced by:  pwfseqlem4  9052
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