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Theorem pwfseqlem4a 8283
Description: Lemma for pwfseqlem4 8284. (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 7353 . . 3  |-  ( a  e.  Fin  <->  a  ~<  om )
2 simpr 447 . . . . . . 7  |-  ( (
ph  /\  a  e.  Fin )  ->  a  e. 
Fin )
3 vex 2791 . . . . . . 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 8281 . . . . . . 7  |-  ( ( a  e.  Fin  /\  s  e.  _V )  ->  ( a F s )  =  ( H `
 ( card `  a
) ) )
122, 3, 11sylancl 643 . . . . . 6  |-  ( (
ph  /\  a  e.  Fin )  ->  ( a F s )  =  ( H `  ( card `  a ) ) )
13 f1of 5472 . . . . . . . . 9  |-  ( H : om -1-1-onto-> X  ->  H : om
--> X )
146, 13syl 15 . . . . . . . 8  |-  ( ph  ->  H : om --> X )
15 fss 5397 . . . . . . . 8  |-  ( ( H : om --> X  /\  X  C_  A )  ->  H : om --> A )
1614, 5, 15syl2anc 642 . . . . . . 7  |-  ( ph  ->  H : om --> A )
17 ficardom 7594 . . . . . . 7  |-  ( a  e.  Fin  ->  ( card `  a )  e. 
om )
18 ffvelrn 5663 . . . . . . 7  |-  ( ( H : om --> A  /\  ( card `  a )  e.  om )  ->  ( H `  ( card `  a ) )  e.  A )
1916, 17, 18syl2an 463 . . . . . 6  |-  ( (
ph  /\  a  e.  Fin )  ->  ( H `
 ( card `  a
) )  e.  A
)
2012, 19eqeltrd 2357 . . . . 5  |-  ( (
ph  /\  a  e.  Fin )  ->  ( a F s )  e.  A )
2120ex 423 . . . 4  |-  ( ph  ->  ( a  e.  Fin  ->  ( a F s )  e.  A ) )
2221adantr 451 . . 3  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( a  e.  Fin  ->  ( a F s )  e.  A ) )
231, 22syl5bir 209 . 2  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( a  ~<  om  ->  ( a F s )  e.  A ) )
24 omelon 7347 . . . . 5  |-  om  e.  On
25 onenon 7582 . . . . 5  |-  ( om  e.  On  ->  om  e.  dom  card )
2624, 25ax-mp 8 . . . 4  |-  om  e.  dom  card
27 simpr3 963 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
s  We  a )
28 19.8a 1718 . . . . . 6  |-  ( s  We  a  ->  E. s 
s  We  a )
2927, 28syl 15 . . . . 5  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  ->  E. s  s  We  a )
30 ween 7662 . . . . 5  |-  ( a  e.  dom  card  <->  E. s 
s  We  a )
3129, 30sylibr 203 . . . 4  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
a  e.  dom  card )
32 domtri2 7622 . . . 4  |-  ( ( om  e.  dom  card  /\  a  e.  dom  card )  ->  ( om  ~<_  a  <->  -.  a  ~<  om ) )
3326, 31, 32sylancr 644 . . 3  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( om  ~<_  a  <->  -.  a  ~<  om ) )
34 nfv 1605 . . . . . . 7  |-  F/ r ( ph  /\  (
( a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  /\  om  ~<_  a ) )
35 nfcv 2419 . . . . . . . . 9  |-  F/_ r
a
36 nfmpt22 5915 . . . . . . . . . 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 2416 . . . . . . . . 9  |-  F/_ r F
38 nfcv 2419 . . . . . . . . 9  |-  F/_ r
s
3935, 37, 38nfov 5881 . . . . . . . 8  |-  F/_ r
( a F s )
4039nfel1 2429 . . . . . . 7  |-  F/ r ( a F s )  e.  ( A 
\  a )
4134, 40nfim 1769 . . . . . 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 3199 . . . . . . . . . 10  |-  ( r  =  s  ->  (
r  C_  ( a  X.  a )  <->  s  C_  ( a  X.  a
) ) )
43 weeq1 4381 . . . . . . . . . 10  |-  ( r  =  s  ->  (
r  We  a  <->  s  We  a ) )
4442, 433anbi23d 1255 . . . . . . . . 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 685 . . . . . . . 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 684 . . . . . . 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 5866 . . . . . . . 8  |-  ( r  =  s  ->  (
a F r )  =  ( a F s ) )
4847eleq1d 2349 . . . . . . 7  |-  ( r  =  s  ->  (
( a F r )  e.  ( A 
\  a )  <->  ( a F s )  e.  ( A  \  a
) ) )
4946, 48imbi12d 311 . . . . . 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 1605 . . . . . . . 8  |-  F/ x
( ph  /\  (
( a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  /\  om  ~<_  a ) )
51 nfcv 2419 . . . . . . . . . 10  |-  F/_ x
a
52 nfmpt21 5914 . . . . . . . . . . 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 2416 . . . . . . . . . 10  |-  F/_ x F
54 nfcv 2419 . . . . . . . . . 10  |-  F/_ x
r
5551, 53, 54nfov 5881 . . . . . . . . 9  |-  F/_ x
( a F r )
5655nfel1 2429 . . . . . . . 8  |-  F/ x
( a F r )  e.  ( A 
\  a )
5750, 56nfim 1769 . . . . . . 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 3199 . . . . . . . . . . . 12  |-  ( x  =  a  ->  (
x  C_  A  <->  a  C_  A ) )
59 xpeq12 4708 . . . . . . . . . . . . . 14  |-  ( ( x  =  a  /\  x  =  a )  ->  ( x  X.  x
)  =  ( a  X.  a ) )
6059anidms 626 . . . . . . . . . . . . 13  |-  ( x  =  a  ->  (
x  X.  x )  =  ( a  X.  a ) )
6160sseq2d 3206 . . . . . . . . . . . 12  |-  ( x  =  a  ->  (
r  C_  ( x  X.  x )  <->  r  C_  ( a  X.  a
) ) )
62 weeq2 4382 . . . . . . . . . . . 12  |-  ( x  =  a  ->  (
r  We  x  <->  r  We  a ) )
6358, 61, 623anbi123d 1252 . . . . . . . . . . 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 4027 . . . . . . . . . . 11  |-  ( x  =  a  ->  ( om 
~<_  x  <->  om  ~<_  a ) )
6563, 64anbi12d 691 . . . . . . . . . 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 248 . . . . . . . . 9  |-  ( x  =  a  ->  ( ps 
<->  ( ( a  C_  A  /\  r  C_  (
a  X.  a )  /\  r  We  a
)  /\  om  ~<_  a ) ) )
6766anbi2d 684 . . . . . . . 8  |-  ( x  =  a  ->  (
( ph  /\  ps )  <->  (
ph  /\  ( (
a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  /\  om  ~<_  a ) ) ) )
68 oveq1 5865 . . . . . . . . 9  |-  ( x  =  a  ->  (
x F r )  =  ( a F r ) )
69 difeq2 3288 . . . . . . . . 9  |-  ( x  =  a  ->  ( A  \  x )  =  ( A  \  a
) )
7068, 69eleq12d 2351 . . . . . . . 8  |-  ( x  =  a  ->  (
( x F r )  e.  ( A 
\  x )  <->  ( a F r )  e.  ( A  \  a
) ) )
7167, 70imbi12d 311 . . . . . . 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 8282 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  ( x F r )  e.  ( A 
\  x ) )
7357, 71, 72chvar 1926 . . . . . 6  |-  ( (
ph  /\  ( (
a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  /\  om  ~<_  a ) )  -> 
( a F r )  e.  ( A 
\  a ) )
7441, 49, 73chvar 1926 . . . . 5  |-  ( (
ph  /\  ( (
a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  /\  om  ~<_  a ) )  -> 
( a F s )  e.  ( A 
\  a ) )
75 eldifi 3298 . . . . 5  |-  ( ( a F s )  e.  ( A  \ 
a )  ->  (
a F s )  e.  A )
7674, 75syl 15 . . . 4  |-  ( (
ph  /\  ( (
a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  /\  om  ~<_  a ) )  -> 
( a F s )  e.  A )
7776expr 598 . . 3  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( om  ~<_  a  -> 
( a F s )  e.  A ) )
7833, 77sylbird 226 . 2  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( -.  a  ~<  om  ->  ( a F s )  e.  A
) )
7923, 78pm2.61d 150 1  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( a F s )  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   ifcif 3565   ~Pcpw 3625   |^|cint 3862   U_ciun 3905   class class class wbr 4023    We wwe 4351   Oncon0 4392   omcom 4656    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    ^m cmap 6772    ~<_ cdom 6861    ~< csdm 6862   Fincfn 6863   cardccrd 7568
This theorem is referenced by:  pwfseqlem4  8284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572
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