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Theorem rankwflemb 7711
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 4010 . . 3  |-  ( A  e.  U. ( R1
" On )  <->  E. y
( A  e.  y  /\  y  e.  ( R1 " On ) ) )
2 r1funlim 7684 . . . . . . . 8  |-  ( Fun 
R1  /\  Lim  dom  R1 )
32simpli 445 . . . . . . 7  |-  Fun  R1
4 fvelima 5770 . . . . . . 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 2496 . . . . . . . . 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 7694 . . . . . . . . . . . 12  |-  Tr  ( R1 `  x )
9 trss 4303 . . . . . . . . . . . 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 3798 . . . . . . . . . . 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 5749 . . . . . . . . . . 11  |-  ( A  e.  ( R1 `  x )  ->  x  e.  dom  R1 )
14 r1sucg 7687 . . . . . . . . . . 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 2512 . . . . . . . . 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 2808 . . . . . 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 1644 . . 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 5749 . . . . . 6  |-  ( A  e.  ( R1 `  suc  x )  ->  suc  x  e.  dom  R1 )
25 fvelrn 5858 . . . . . 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 4883 . . . . . 6  |-  ( R1
" On )  =  ran  ( R1  |`  On )
28 funrel 5463 . . . . . . . . 9  |-  ( Fun 
R1  ->  Rel  R1 )
293, 28ax-mp 8 . . . . . . . 8  |-  Rel  R1
302simpri 449 . . . . . . . . 9  |-  Lim  dom  R1
31 limord 4632 . . . . . . . . 9  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
32 ordsson 4762 . . . . . . . . 9  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
3330, 31, 32mp2b 10 . . . . . . . 8  |-  dom  R1  C_  On
34 relssres 5175 . . . . . . . 8  |-  ( ( Rel  R1  /\  dom  R1  C_  On )  ->  ( R1  |`  On )  =  R1 )
3529, 33, 34mp2an 654 . . . . . . 7  |-  ( R1  |`  On )  =  R1
3635rneqi 5088 . . . . . 6  |-  ran  ( R1  |`  On )  =  ran  R1
3727, 36eqtri 2455 . . . . 5  |-  ( R1
" On )  =  ran  R1
3826, 37syl6eleqr 2526 . . . 4  |-  ( A  e.  ( R1 `  suc  x )  ->  ( R1 `  suc  x )  e.  ( R1 " On ) )
39 elunii 4012 . . . 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 2818 . 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 1550    = wceq 1652    e. wcel 1725   E.wrex 2698    C_ wss 3312   ~Pcpw 3791   U.cuni 4007   Tr wtr 4294   Ord word 4572   Oncon0 4573   Lim wlim 4574   suc csuc 4575   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   Rel wrel 4875   Fun wfun 5440   ` cfv 5446   R1cr1 7680
This theorem is referenced by:  rankf  7712  r1elwf  7714  rankvalb  7715  rankidb  7718  rankwflem  7733  tcrank  7800  dfac12r  8018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-recs 6625  df-rdg 6660  df-r1 7682
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