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Theorem rankwflemb 7465
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 3830 . . 3  |-  ( A  e.  U. ( R1
" On )  <->  E. y
( A  e.  y  /\  y  e.  ( R1 " On ) ) )
2 r1funlim 7438 . . . . . . . 8  |-  ( Fun 
R1  /\  Lim  dom  R1 )
32simpli 444 . . . . . . 7  |-  Fun  R1
4 fvelima 5574 . . . . . . 7  |-  ( ( Fun  R1  /\  y  e.  ( R1 " On ) )  ->  E. x  e.  On  ( R1 `  x )  =  y )
53, 4mpan 651 . . . . . 6  |-  ( y  e.  ( R1 " On )  ->  E. x  e.  On  ( R1 `  x )  =  y )
6 eleq2 2344 . . . . . . . . 9  |-  ( ( R1 `  x )  =  y  ->  ( A  e.  ( R1 `  x )  <->  A  e.  y ) )
76biimprcd 216 . . . . . . . 8  |-  ( A  e.  y  ->  (
( R1 `  x
)  =  y  ->  A  e.  ( R1 `  x ) ) )
8 r1tr 7448 . . . . . . . . . . . 12  |-  Tr  ( R1 `  x )
9 trss 4122 . . . . . . . . . . . 12  |-  ( Tr  ( R1 `  x
)  ->  ( A  e.  ( R1 `  x
)  ->  A  C_  ( R1 `  x ) ) )
108, 9ax-mp 8 . . . . . . . . . . 11  |-  ( A  e.  ( R1 `  x )  ->  A  C_  ( R1 `  x
) )
11 elpwg 3632 . . . . . . . . . . 11  |-  ( A  e.  ( R1 `  x )  ->  ( A  e.  ~P ( R1 `  x )  <->  A  C_  ( R1 `  x ) ) )
1210, 11mpbird 223 . . . . . . . . . 10  |-  ( A  e.  ( R1 `  x )  ->  A  e.  ~P ( R1 `  x ) )
13 elfvdm 5554 . . . . . . . . . . 11  |-  ( A  e.  ( R1 `  x )  ->  x  e.  dom  R1 )
14 r1sucg 7441 . . . . . . . . . . 11  |-  ( x  e.  dom  R1  ->  ( R1 `  suc  x
)  =  ~P ( R1 `  x ) )
1513, 14syl 15 . . . . . . . . . 10  |-  ( A  e.  ( R1 `  x )  ->  ( R1 `  suc  x )  =  ~P ( R1
`  x ) )
1612, 15eleqtrrd 2360 . . . . . . . . 9  |-  ( A  e.  ( R1 `  x )  ->  A  e.  ( R1 `  suc  x ) )
1716a1i 10 . . . . . . . 8  |-  ( x  e.  On  ->  ( A  e.  ( R1 `  x )  ->  A  e.  ( R1 `  suc  x ) ) )
187, 17syl9 66 . . . . . . 7  |-  ( A  e.  y  ->  (
x  e.  On  ->  ( ( R1 `  x
)  =  y  ->  A  e.  ( R1 ` 
suc  x ) ) ) )
1918reximdvai 2653 . . . . . 6  |-  ( A  e.  y  ->  ( E. x  e.  On  ( R1 `  x )  =  y  ->  E. x  e.  On  A  e.  ( R1 `  suc  x
) ) )
205, 19syl5 28 . . . . 5  |-  ( A  e.  y  ->  (
y  e.  ( R1
" On )  ->  E. x  e.  On  A  e.  ( R1 ` 
suc  x ) ) )
2120imp 418 . . . 4  |-  ( ( A  e.  y  /\  y  e.  ( R1 " On ) )  ->  E. x  e.  On  A  e.  ( R1 ` 
suc  x ) )
2221exlimiv 1666 . . 3  |-  ( E. y ( A  e.  y  /\  y  e.  ( R1 " On ) )  ->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
231, 22sylbi 187 . 2  |-  ( A  e.  U. ( R1
" On )  ->  E. x  e.  On  A  e.  ( R1 ` 
suc  x ) )
24 elfvdm 5554 . . . . . 6  |-  ( A  e.  ( R1 `  suc  x )  ->  suc  x  e.  dom  R1 )
25 fvelrn 5661 . . . . . 6  |-  ( ( Fun  R1  /\  suc  x  e.  dom  R1 )  ->  ( R1 `  suc  x )  e.  ran  R1 )
263, 24, 25sylancr 644 . . . . 5  |-  ( A  e.  ( R1 `  suc  x )  ->  ( R1 `  suc  x )  e.  ran  R1 )
27 df-ima 4702 . . . . . 6  |-  ( R1
" On )  =  ran  ( R1  |`  On )
28 funrel 5272 . . . . . . . . 9  |-  ( Fun 
R1  ->  Rel  R1 )
293, 28ax-mp 8 . . . . . . . 8  |-  Rel  R1
302simpri 448 . . . . . . . . 9  |-  Lim  dom  R1
31 limord 4451 . . . . . . . . 9  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
32 ordsson 4581 . . . . . . . . 9  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
3330, 31, 32mp2b 9 . . . . . . . 8  |-  dom  R1  C_  On
34 relssres 4992 . . . . . . . 8  |-  ( ( Rel  R1  /\  dom  R1  C_  On )  ->  ( R1  |`  On )  =  R1 )
3529, 33, 34mp2an 653 . . . . . . 7  |-  ( R1  |`  On )  =  R1
3635rneqi 4905 . . . . . 6  |-  ran  ( R1  |`  On )  =  ran  R1
3727, 36eqtri 2303 . . . . 5  |-  ( R1
" On )  =  ran  R1
3826, 37syl6eleqr 2374 . . . 4  |-  ( A  e.  ( R1 `  suc  x )  ->  ( R1 `  suc  x )  e.  ( R1 " On ) )
39 elunii 3832 . . . 4  |-  ( ( A  e.  ( R1
`  suc  x )  /\  ( R1 `  suc  x )  e.  ( R1 " On ) )  ->  A  e.  U. ( R1 " On ) )
4038, 39mpdan 649 . . 3  |-  ( A  e.  ( R1 `  suc  x )  ->  A  e.  U. ( R1 " On ) )
4140rexlimivw 2663 . 2  |-  ( E. x  e.  On  A  e.  ( R1 `  suc  x )  ->  A  e.  U. ( R1 " On ) )
4223, 41impbii 180 1  |-  ( A  e.  U. ( R1
" On )  <->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   Tr wtr 4113   Ord word 4391   Oncon0 4392   Lim wlim 4393   suc csuc 4394   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Rel wrel 4694   Fun wfun 5249   ` cfv 5255   R1cr1 7434
This theorem is referenced by:  rankf  7466  r1elwf  7468  rankvalb  7469  rankidb  7472  rankwflem  7487  tcrank  7554  dfac12r  7772
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-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-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-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-recs 6388  df-rdg 6423  df-r1 7436
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