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Theorem rankwflemb 8272
Description: Two ways of saying a set is well-founded. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
rankwflemb  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
Distinct variable group:    x, A

Proof of Theorem rankwflemb
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eluni 4222 . . 3  |-  ( A  e.  U. ( R1
" On )  <->  E. y
( A  e.  y  /\  y  e.  ( R1 " On ) ) )
2 r1funlim 8245 . . . . . . . 8  |-  ( Fun 
R1  /\  Lim  dom  R1 )
32simpli 459 . . . . . . 7  |-  Fun  R1
4 fvelima 5933 . . . . . . 7  |-  ( ( Fun  R1  /\  y  e.  ( R1 " On ) )  ->  E. x  e.  On  ( R1 `  x )  =  y )
53, 4mpan 674 . . . . . 6  |-  ( y  e.  ( R1 " On )  ->  E. x  e.  On  ( R1 `  x )  =  y )
6 eleq2 2496 . . . . . . . . 9  |-  ( ( R1 `  x )  =  y  ->  ( A  e.  ( R1 `  x )  <->  A  e.  y ) )
76biimprcd 228 . . . . . . . 8  |-  ( A  e.  y  ->  (
( R1 `  x
)  =  y  ->  A  e.  ( R1 `  x ) ) )
8 r1tr 8255 . . . . . . . . . . . 12  |-  Tr  ( R1 `  x )
9 trss 4527 . . . . . . . . . . . 12  |-  ( Tr  ( R1 `  x
)  ->  ( A  e.  ( R1 `  x
)  ->  A  C_  ( R1 `  x ) ) )
108, 9ax-mp 5 . . . . . . . . . . 11  |-  ( A  e.  ( R1 `  x )  ->  A  C_  ( R1 `  x
) )
11 elpwg 3989 . . . . . . . . . . 11  |-  ( A  e.  ( R1 `  x )  ->  ( A  e.  ~P ( R1 `  x )  <->  A  C_  ( R1 `  x ) ) )
1210, 11mpbird 235 . . . . . . . . . 10  |-  ( A  e.  ( R1 `  x )  ->  A  e.  ~P ( R1 `  x ) )
13 elfvdm 5907 . . . . . . . . . . 11  |-  ( A  e.  ( R1 `  x )  ->  x  e.  dom  R1 )
14 r1sucg 8248 . . . . . . . . . . 11  |-  ( x  e.  dom  R1  ->  ( R1 `  suc  x
)  =  ~P ( R1 `  x ) )
1513, 14syl 17 . . . . . . . . . 10  |-  ( A  e.  ( R1 `  x )  ->  ( R1 `  suc  x )  =  ~P ( R1
`  x ) )
1612, 15eleqtrrd 2510 . . . . . . . . 9  |-  ( A  e.  ( R1 `  x )  ->  A  e.  ( R1 `  suc  x ) )
1716a1i 11 . . . . . . . 8  |-  ( x  e.  On  ->  ( A  e.  ( R1 `  x )  ->  A  e.  ( R1 `  suc  x ) ) )
187, 17syl9 73 . . . . . . 7  |-  ( A  e.  y  ->  (
x  e.  On  ->  ( ( R1 `  x
)  =  y  ->  A  e.  ( R1 ` 
suc  x ) ) ) )
1918reximdvai 2894 . . . . . 6  |-  ( A  e.  y  ->  ( E. x  e.  On  ( R1 `  x )  =  y  ->  E. x  e.  On  A  e.  ( R1 `  suc  x
) ) )
205, 19syl5 33 . . . . 5  |-  ( A  e.  y  ->  (
y  e.  ( R1
" On )  ->  E. x  e.  On  A  e.  ( R1 ` 
suc  x ) ) )
2120imp 430 . . . 4  |-  ( ( A  e.  y  /\  y  e.  ( R1 " On ) )  ->  E. x  e.  On  A  e.  ( R1 ` 
suc  x ) )
2221exlimiv 1770 . . 3  |-  ( E. y ( A  e.  y  /\  y  e.  ( R1 " On ) )  ->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
231, 22sylbi 198 . 2  |-  ( A  e.  U. ( R1
" On )  ->  E. x  e.  On  A  e.  ( R1 ` 
suc  x ) )
24 elfvdm 5907 . . . . . 6  |-  ( A  e.  ( R1 `  suc  x )  ->  suc  x  e.  dom  R1 )
25 fvelrn 6030 . . . . . 6  |-  ( ( Fun  R1  /\  suc  x  e.  dom  R1 )  ->  ( R1 `  suc  x )  e.  ran  R1 )
263, 24, 25sylancr 667 . . . . 5  |-  ( A  e.  ( R1 `  suc  x )  ->  ( R1 `  suc  x )  e.  ran  R1 )
27 df-ima 4866 . . . . . 6  |-  ( R1
" On )  =  ran  ( R1  |`  On )
28 funrel 5618 . . . . . . . . 9  |-  ( Fun 
R1  ->  Rel  R1 )
293, 28ax-mp 5 . . . . . . . 8  |-  Rel  R1
302simpri 463 . . . . . . . . 9  |-  Lim  dom  R1
31 limord 5501 . . . . . . . . 9  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
32 ordsson 6630 . . . . . . . . 9  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
3330, 31, 32mp2b 10 . . . . . . . 8  |-  dom  R1  C_  On
34 relssres 5161 . . . . . . . 8  |-  ( ( Rel  R1  /\  dom  R1  C_  On )  ->  ( R1  |`  On )  =  R1 )
3529, 33, 34mp2an 676 . . . . . . 7  |-  ( R1  |`  On )  =  R1
3635rneqi 5080 . . . . . 6  |-  ran  ( R1  |`  On )  =  ran  R1
3727, 36eqtri 2451 . . . . 5  |-  ( R1
" On )  =  ran  R1
3826, 37syl6eleqr 2518 . . . 4  |-  ( A  e.  ( R1 `  suc  x )  ->  ( R1 `  suc  x )  e.  ( R1 " On ) )
39 elunii 4224 . . . 4  |-  ( ( A  e.  ( R1
`  suc  x )  /\  ( R1 `  suc  x )  e.  ( R1 " On ) )  ->  A  e.  U. ( R1 " On ) )
4038, 39mpdan 672 . . 3  |-  ( A  e.  ( R1 `  suc  x )  ->  A  e.  U. ( R1 " On ) )
4140rexlimivw 2911 . 2  |-  ( E. x  e.  On  A  e.  ( R1 `  suc  x )  ->  A  e.  U. ( R1 " On ) )
4223, 41impbii 190 1  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   E.wrex 2772    C_ wss 3436   ~Pcpw 3981   U.cuni 4219   Tr wtr 4518   dom cdm 4853   ran crn 4854    |` cres 4855   "cima 4856   Rel wrel 4858   Ord word 5441   Oncon0 5442   Lim wlim 5443   suc csuc 5444   Fun wfun 5595   ` cfv 5601   R1cr1 8241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-r1 8243
This theorem is referenced by:  rankf  8273  r1elwf  8275  rankvalb  8276  rankidb  8279  rankwflem  8294  tcrank  8363  dfac12r  8583
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