MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwfseqlem4a Structured version   Unicode version

Theorem pwfseqlem4a 9056
Description: Lemma for pwfseqlem4 9057. (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 8086 . . 3  |-  ( a  e.  Fin  <->  a  ~<  om )
2 simpr 461 . . . . . . 7  |-  ( (
ph  /\  a  e.  Fin )  ->  a  e. 
Fin )
3 vex 3112 . . . . . . 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 9054 . . . . . . 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 )
1514, 5fssd 5746 . . . . . . 7  |-  ( ph  ->  H : om --> A )
16 ficardom 8359 . . . . . . 7  |-  ( a  e.  Fin  ->  ( card `  a )  e. 
om )
17 ffvelrn 6030 . . . . . . 7  |-  ( ( H : om --> A  /\  ( card `  a )  e.  om )  ->  ( H `  ( card `  a ) )  e.  A )
1815, 16, 17syl2an 477 . . . . . 6  |-  ( (
ph  /\  a  e.  Fin )  ->  ( H `
 ( card `  a
) )  e.  A
)
1912, 18eqeltrd 2545 . . . . 5  |-  ( (
ph  /\  a  e.  Fin )  ->  ( a F s )  e.  A )
2019ex 434 . . . 4  |-  ( ph  ->  ( a  e.  Fin  ->  ( a F s )  e.  A ) )
2120adantr 465 . . 3  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( a  e.  Fin  ->  ( a F s )  e.  A ) )
221, 21syl5bir 218 . 2  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( a  ~<  om  ->  ( a F s )  e.  A ) )
23 omelon 8080 . . . . 5  |-  om  e.  On
24 onenon 8347 . . . . 5  |-  ( om  e.  On  ->  om  e.  dom  card )
2523, 24ax-mp 5 . . . 4  |-  om  e.  dom  card
26 simpr3 1004 . . . . . 6  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
s  We  a )
27 19.8a 1858 . . . . . 6  |-  ( s  We  a  ->  E. s 
s  We  a )
2826, 27syl 16 . . . . 5  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  ->  E. s  s  We  a )
29 ween 8433 . . . . 5  |-  ( a  e.  dom  card  <->  E. s 
s  We  a )
3028, 29sylibr 212 . . . 4  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
a  e.  dom  card )
31 domtri2 8387 . . . 4  |-  ( ( om  e.  dom  card  /\  a  e.  dom  card )  ->  ( om  ~<_  a  <->  -.  a  ~<  om ) )
3225, 30, 31sylancr 663 . . 3  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( om  ~<_  a  <->  -.  a  ~<  om ) )
33 nfv 1708 . . . . . . 7  |-  F/ r ( ph  /\  (
( a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  /\  om  ~<_  a ) )
34 nfcv 2619 . . . . . . . . 9  |-  F/_ r
a
35 nfmpt22 6364 . . . . . . . . . 10  |-  F/_ r
( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
3610, 35nfcxfr 2617 . . . . . . . . 9  |-  F/_ r F
37 nfcv 2619 . . . . . . . . 9  |-  F/_ r
s
3834, 36, 37nfov 6322 . . . . . . . 8  |-  F/_ r
( a F s )
3938nfel1 2635 . . . . . . 7  |-  F/ r ( a F s )  e.  ( A 
\  a )
4033, 39nfim 1921 . . . . . 6  |-  F/ r ( ( ph  /\  ( ( a  C_  A  /\  s  C_  (
a  X.  a )  /\  s  We  a
)  /\  om  ~<_  a ) )  ->  ( a F s )  e.  ( A  \  a
) )
41 sseq1 3520 . . . . . . . . . 10  |-  ( r  =  s  ->  (
r  C_  ( a  X.  a )  <->  s  C_  ( a  X.  a
) ) )
42 weeq1 4876 . . . . . . . . . 10  |-  ( r  =  s  ->  (
r  We  a  <->  s  We  a ) )
4341, 423anbi23d 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 ) ) )
4443anbi1d 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 ) ) )
4544anbi2d 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 ) ) ) )
46 oveq2 6304 . . . . . . . 8  |-  ( r  =  s  ->  (
a F r )  =  ( a F s ) )
4746eleq1d 2526 . . . . . . 7  |-  ( r  =  s  ->  (
( a F r )  e.  ( A 
\  a )  <->  ( a F s )  e.  ( A  \  a
) ) )
4845, 47imbi12d 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 ) ) ) )
49 nfv 1708 . . . . . . . 8  |-  F/ x
( ph  /\  (
( a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  /\  om  ~<_  a ) )
50 nfcv 2619 . . . . . . . . . 10  |-  F/_ x
a
51 nfmpt21 6363 . . . . . . . . . . 11  |-  F/_ x
( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
5210, 51nfcxfr 2617 . . . . . . . . . 10  |-  F/_ x F
53 nfcv 2619 . . . . . . . . . 10  |-  F/_ x
r
5450, 52, 53nfov 6322 . . . . . . . . 9  |-  F/_ x
( a F r )
5554nfel1 2635 . . . . . . . 8  |-  F/ x
( a F r )  e.  ( A 
\  a )
5649, 55nfim 1921 . . . . . . 7  |-  F/ x
( ( ph  /\  ( ( a  C_  A  /\  r  C_  (
a  X.  a )  /\  r  We  a
)  /\  om  ~<_  a ) )  ->  ( a F r )  e.  ( A  \  a
) )
57 sseq1 3520 . . . . . . . . . . . 12  |-  ( x  =  a  ->  (
x  C_  A  <->  a  C_  A ) )
58 xpeq12 5027 . . . . . . . . . . . . . 14  |-  ( ( x  =  a  /\  x  =  a )  ->  ( x  X.  x
)  =  ( a  X.  a ) )
5958anidms 645 . . . . . . . . . . . . 13  |-  ( x  =  a  ->  (
x  X.  x )  =  ( a  X.  a ) )
6059sseq2d 3527 . . . . . . . . . . . 12  |-  ( x  =  a  ->  (
r  C_  ( x  X.  x )  <->  r  C_  ( a  X.  a
) ) )
61 weeq2 4877 . . . . . . . . . . . 12  |-  ( x  =  a  ->  (
r  We  x  <->  r  We  a ) )
6257, 60, 613anbi123d 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 ) ) )
63 breq2 4460 . . . . . . . . . . 11  |-  ( x  =  a  ->  ( om 
~<_  x  <->  om  ~<_  a ) )
6462, 63anbi12d 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 ) ) )
657, 64syl5bb 257 . . . . . . . . 9  |-  ( x  =  a  ->  ( ps 
<->  ( ( a  C_  A  /\  r  C_  (
a  X.  a )  /\  r  We  a
)  /\  om  ~<_  a ) ) )
6665anbi2d 703 . . . . . . . 8  |-  ( x  =  a  ->  (
( ph  /\  ps )  <->  (
ph  /\  ( (
a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  /\  om  ~<_  a ) ) ) )
67 oveq1 6303 . . . . . . . . 9  |-  ( x  =  a  ->  (
x F r )  =  ( a F r ) )
68 difeq2 3612 . . . . . . . . 9  |-  ( x  =  a  ->  ( A  \  x )  =  ( A  \  a
) )
6967, 68eleq12d 2539 . . . . . . . 8  |-  ( x  =  a  ->  (
( x F r )  e.  ( A 
\  x )  <->  ( a F r )  e.  ( A  \  a
) ) )
7066, 69imbi12d 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 ) ) ) )
714, 5, 6, 7, 8, 9, 10pwfseqlem3 9055 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  ( x F r )  e.  ( A 
\  x ) )
7256, 70, 71chvar 2014 . . . . . 6  |-  ( (
ph  /\  ( (
a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  /\  om  ~<_  a ) )  -> 
( a F r )  e.  ( A 
\  a ) )
7340, 48, 72chvar 2014 . . . . 5  |-  ( (
ph  /\  ( (
a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  /\  om  ~<_  a ) )  -> 
( a F s )  e.  ( A 
\  a ) )
7473eldifad 3483 . . . 4  |-  ( (
ph  /\  ( (
a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  /\  om  ~<_  a ) )  -> 
( a F s )  e.  A )
7574expr 615 . . 3  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( om  ~<_  a  -> 
( a F s )  e.  A ) )
7632, 75sylbird 235 . 2  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( -.  a  ~<  om  ->  ( a F s )  e.  A
) )
7722, 76pm2.61d 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 1395   E.wex 1613    e. wcel 1819   {crab 2811   _Vcvv 3109    \ cdif 3468    C_ wss 3471   ifcif 3944   ~Pcpw 4015   |^|cint 4288   U_ciun 4332   class class class wbr 4456    We wwe 4846   Oncon0 4887    X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009   -->wf 5590   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   omcom 6699    ^m cmap 7438    ~<_ cdom 7533    ~< csdm 7534   Fincfn 7535   cardccrd 8333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337
This theorem is referenced by:  pwfseqlem4  9057
  Copyright terms: Public domain W3C validator