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Theorem rankwflemb 8212
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 4248 . . 3  |-  ( A  e.  U. ( R1
" On )  <->  E. y
( A  e.  y  /\  y  e.  ( R1 " On ) ) )
2 r1funlim 8185 . . . . . . . 8  |-  ( Fun 
R1  /\  Lim  dom  R1 )
32simpli 458 . . . . . . 7  |-  Fun  R1
4 fvelima 5920 . . . . . . 7  |-  ( ( Fun  R1  /\  y  e.  ( R1 " On ) )  ->  E. x  e.  On  ( R1 `  x )  =  y )
53, 4mpan 670 . . . . . 6  |-  ( y  e.  ( R1 " On )  ->  E. x  e.  On  ( R1 `  x )  =  y )
6 eleq2 2540 . . . . . . . . 9  |-  ( ( R1 `  x )  =  y  ->  ( A  e.  ( R1 `  x )  <->  A  e.  y ) )
76biimprcd 225 . . . . . . . 8  |-  ( A  e.  y  ->  (
( R1 `  x
)  =  y  ->  A  e.  ( R1 `  x ) ) )
8 r1tr 8195 . . . . . . . . . . . 12  |-  Tr  ( R1 `  x )
9 trss 4549 . . . . . . . . . . . 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 4018 . . . . . . . . . . 11  |-  ( A  e.  ( R1 `  x )  ->  ( A  e.  ~P ( R1 `  x )  <->  A  C_  ( R1 `  x ) ) )
1210, 11mpbird 232 . . . . . . . . . 10  |-  ( A  e.  ( R1 `  x )  ->  A  e.  ~P ( R1 `  x ) )
13 elfvdm 5892 . . . . . . . . . . 11  |-  ( A  e.  ( R1 `  x )  ->  x  e.  dom  R1 )
14 r1sucg 8188 . . . . . . . . . . 11  |-  ( x  e.  dom  R1  ->  ( R1 `  suc  x
)  =  ~P ( R1 `  x ) )
1513, 14syl 16 . . . . . . . . . 10  |-  ( A  e.  ( R1 `  x )  ->  ( R1 `  suc  x )  =  ~P ( R1
`  x ) )
1612, 15eleqtrrd 2558 . . . . . . . . 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 71 . . . . . . 7  |-  ( A  e.  y  ->  (
x  e.  On  ->  ( ( R1 `  x
)  =  y  ->  A  e.  ( R1 ` 
suc  x ) ) ) )
1918reximdvai 2935 . . . . . 6  |-  ( A  e.  y  ->  ( E. x  e.  On  ( R1 `  x )  =  y  ->  E. x  e.  On  A  e.  ( R1 `  suc  x
) ) )
205, 19syl5 32 . . . . 5  |-  ( A  e.  y  ->  (
y  e.  ( R1
" On )  ->  E. x  e.  On  A  e.  ( R1 ` 
suc  x ) ) )
2120imp 429 . . . 4  |-  ( ( A  e.  y  /\  y  e.  ( R1 " On ) )  ->  E. x  e.  On  A  e.  ( R1 ` 
suc  x ) )
2221exlimiv 1698 . . 3  |-  ( E. y ( A  e.  y  /\  y  e.  ( R1 " On ) )  ->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
231, 22sylbi 195 . 2  |-  ( A  e.  U. ( R1
" On )  ->  E. x  e.  On  A  e.  ( R1 ` 
suc  x ) )
24 elfvdm 5892 . . . . . 6  |-  ( A  e.  ( R1 `  suc  x )  ->  suc  x  e.  dom  R1 )
25 fvelrn 6018 . . . . . 6  |-  ( ( Fun  R1  /\  suc  x  e.  dom  R1 )  ->  ( R1 `  suc  x )  e.  ran  R1 )
263, 24, 25sylancr 663 . . . . 5  |-  ( A  e.  ( R1 `  suc  x )  ->  ( R1 `  suc  x )  e.  ran  R1 )
27 df-ima 5012 . . . . . 6  |-  ( R1
" On )  =  ran  ( R1  |`  On )
28 funrel 5605 . . . . . . . . 9  |-  ( Fun 
R1  ->  Rel  R1 )
293, 28ax-mp 5 . . . . . . . 8  |-  Rel  R1
302simpri 462 . . . . . . . . 9  |-  Lim  dom  R1
31 limord 4937 . . . . . . . . 9  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
32 ordsson 6610 . . . . . . . . 9  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
3330, 31, 32mp2b 10 . . . . . . . 8  |-  dom  R1  C_  On
34 relssres 5311 . . . . . . . 8  |-  ( ( Rel  R1  /\  dom  R1  C_  On )  ->  ( R1  |`  On )  =  R1 )
3529, 33, 34mp2an 672 . . . . . . 7  |-  ( R1  |`  On )  =  R1
3635rneqi 5229 . . . . . 6  |-  ran  ( R1  |`  On )  =  ran  R1
3727, 36eqtri 2496 . . . . 5  |-  ( R1
" On )  =  ran  R1
3826, 37syl6eleqr 2566 . . . 4  |-  ( A  e.  ( R1 `  suc  x )  ->  ( R1 `  suc  x )  e.  ( R1 " On ) )
39 elunii 4250 . . . 4  |-  ( ( A  e.  ( R1
`  suc  x )  /\  ( R1 `  suc  x )  e.  ( R1 " On ) )  ->  A  e.  U. ( R1 " On ) )
4038, 39mpdan 668 . . 3  |-  ( A  e.  ( R1 `  suc  x )  ->  A  e.  U. ( R1 " On ) )
4140rexlimivw 2952 . 2  |-  ( E. x  e.  On  A  e.  ( R1 `  suc  x )  ->  A  e.  U. ( R1 " On ) )
4223, 41impbii 188 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 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   E.wrex 2815    C_ wss 3476   ~Pcpw 4010   U.cuni 4245   Tr wtr 4540   Ord word 4877   Oncon0 4878   Lim wlim 4879   suc csuc 4880   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002   Rel wrel 5004   Fun wfun 5582   ` cfv 5588   R1cr1 8181
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6686  df-recs 7043  df-rdg 7077  df-r1 8183
This theorem is referenced by:  rankf  8213  r1elwf  8215  rankvalb  8216  rankidb  8219  rankwflem  8234  tcrank  8303  dfac12r  8527
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