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Theorem rankwflemb 7645
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 3953 . . 3  |-  ( A  e.  U. ( R1
" On )  <->  E. y
( A  e.  y  /\  y  e.  ( R1 " On ) ) )
2 r1funlim 7618 . . . . . . . 8  |-  ( Fun 
R1  /\  Lim  dom  R1 )
32simpli 445 . . . . . . 7  |-  Fun  R1
4 fvelima 5710 . . . . . . 7  |-  ( ( Fun  R1  /\  y  e.  ( R1 " On ) )  ->  E. x  e.  On  ( R1 `  x )  =  y )
53, 4mpan 652 . . . . . 6  |-  ( y  e.  ( R1 " On )  ->  E. x  e.  On  ( R1 `  x )  =  y )
6 eleq2 2441 . . . . . . . . 9  |-  ( ( R1 `  x )  =  y  ->  ( A  e.  ( R1 `  x )  <->  A  e.  y ) )
76biimprcd 217 . . . . . . . 8  |-  ( A  e.  y  ->  (
( R1 `  x
)  =  y  ->  A  e.  ( R1 `  x ) ) )
8 r1tr 7628 . . . . . . . . . . . 12  |-  Tr  ( R1 `  x )
9 trss 4245 . . . . . . . . . . . 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 3742 . . . . . . . . . . 11  |-  ( A  e.  ( R1 `  x )  ->  ( A  e.  ~P ( R1 `  x )  <->  A  C_  ( R1 `  x ) ) )
1210, 11mpbird 224 . . . . . . . . . 10  |-  ( A  e.  ( R1 `  x )  ->  A  e.  ~P ( R1 `  x ) )
13 elfvdm 5690 . . . . . . . . . . 11  |-  ( A  e.  ( R1 `  x )  ->  x  e.  dom  R1 )
14 r1sucg 7621 . . . . . . . . . . 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 2457 . . . . . . . . 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 68 . . . . . . 7  |-  ( A  e.  y  ->  (
x  e.  On  ->  ( ( R1 `  x
)  =  y  ->  A  e.  ( R1 ` 
suc  x ) ) ) )
1918reximdvai 2752 . . . . . 6  |-  ( A  e.  y  ->  ( E. x  e.  On  ( R1 `  x )  =  y  ->  E. x  e.  On  A  e.  ( R1 `  suc  x
) ) )
205, 19syl5 30 . . . . 5  |-  ( A  e.  y  ->  (
y  e.  ( R1
" On )  ->  E. x  e.  On  A  e.  ( R1 ` 
suc  x ) ) )
2120imp 419 . . . 4  |-  ( ( A  e.  y  /\  y  e.  ( R1 " On ) )  ->  E. x  e.  On  A  e.  ( R1 ` 
suc  x ) )
2221exlimiv 1641 . . 3  |-  ( E. y ( A  e.  y  /\  y  e.  ( R1 " On ) )  ->  E. x  e.  On  A  e.  ( R1 `  suc  x
) )
231, 22sylbi 188 . 2  |-  ( A  e.  U. ( R1
" On )  ->  E. x  e.  On  A  e.  ( R1 ` 
suc  x ) )
24 elfvdm 5690 . . . . . 6  |-  ( A  e.  ( R1 `  suc  x )  ->  suc  x  e.  dom  R1 )
25 fvelrn 5798 . . . . . 6  |-  ( ( Fun  R1  /\  suc  x  e.  dom  R1 )  ->  ( R1 `  suc  x )  e.  ran  R1 )
263, 24, 25sylancr 645 . . . . 5  |-  ( A  e.  ( R1 `  suc  x )  ->  ( R1 `  suc  x )  e.  ran  R1 )
27 df-ima 4824 . . . . . 6  |-  ( R1
" On )  =  ran  ( R1  |`  On )
28 funrel 5404 . . . . . . . . 9  |-  ( Fun 
R1  ->  Rel  R1 )
293, 28ax-mp 8 . . . . . . . 8  |-  Rel  R1
302simpri 449 . . . . . . . . 9  |-  Lim  dom  R1
31 limord 4574 . . . . . . . . 9  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
32 ordsson 4703 . . . . . . . . 9  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
3330, 31, 32mp2b 10 . . . . . . . 8  |-  dom  R1  C_  On
34 relssres 5116 . . . . . . . 8  |-  ( ( Rel  R1  /\  dom  R1  C_  On )  ->  ( R1  |`  On )  =  R1 )
3529, 33, 34mp2an 654 . . . . . . 7  |-  ( R1  |`  On )  =  R1
3635rneqi 5029 . . . . . 6  |-  ran  ( R1  |`  On )  =  ran  R1
3727, 36eqtri 2400 . . . . 5  |-  ( R1
" On )  =  ran  R1
3826, 37syl6eleqr 2471 . . . 4  |-  ( A  e.  ( R1 `  suc  x )  ->  ( R1 `  suc  x )  e.  ( R1 " On ) )
39 elunii 3955 . . . 4  |-  ( ( A  e.  ( R1
`  suc  x )  /\  ( R1 `  suc  x )  e.  ( R1 " On ) )  ->  A  e.  U. ( R1 " On ) )
4038, 39mpdan 650 . . 3  |-  ( A  e.  ( R1 `  suc  x )  ->  A  e.  U. ( R1 " On ) )
4140rexlimivw 2762 . 2  |-  ( E. x  e.  On  A  e.  ( R1 `  suc  x )  ->  A  e.  U. ( R1 " On ) )
4223, 41impbii 181 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 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   E.wrex 2643    C_ wss 3256   ~Pcpw 3735   U.cuni 3950   Tr wtr 4236   Ord word 4514   Oncon0 4515   Lim wlim 4516   suc csuc 4517   dom cdm 4811   ran crn 4812    |` cres 4813   "cima 4814   Rel wrel 4816   Fun wfun 5381   ` cfv 5387   R1cr1 7614
This theorem is referenced by:  rankf  7646  r1elwf  7648  rankvalb  7649  rankidb  7652  rankwflem  7667  tcrank  7734  dfac12r  7952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-recs 6562  df-rdg 6597  df-r1 7616
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