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Theorem rankr1bg 8226
Description: A relationship between rank and  R1. See rankr1ag 8225 for the membership version. (Contributed by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
rankr1bg  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  C_  ( R1 `  B )  <-> 
( rank `  A )  C_  B ) )

Proof of Theorem rankr1bg
StepHypRef Expression
1 r1funlim 8189 . . . . 5  |-  ( Fun 
R1  /\  Lim  dom  R1 )
21simpri 463 . . . 4  |-  Lim  dom  R1
3 limsuc 6634 . . . 4  |-  ( Lim 
dom  R1  ->  ( B  e.  dom  R1  <->  suc  B  e. 
dom  R1 ) )
42, 3ax-mp 5 . . 3  |-  ( B  e.  dom  R1  <->  suc  B  e. 
dom  R1 )
5 rankr1ag 8225 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  suc  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  ( rank `  A
)  e.  suc  B
) )
64, 5sylan2b 477 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  ( rank `  A
)  e.  suc  B
) )
7 r1sucg 8192 . . . . 5  |-  ( B  e.  dom  R1  ->  ( R1 `  suc  B
)  =  ~P ( R1 `  B ) )
87adantl 467 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( R1 `  suc  B )  =  ~P ( R1 `  B ) )
98eleq2d 2491 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  A  e.  ~P ( R1 `  B ) ) )
10 fvex 5835 . . . 4  |-  ( R1
`  B )  e. 
_V
1110elpw2 4531 . . 3  |-  ( A  e.  ~P ( R1
`  B )  <->  A  C_  ( R1 `  B ) )
129, 11syl6rbb 265 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  C_  ( R1 `  B )  <-> 
A  e.  ( R1
`  suc  B )
) )
13 rankon 8218 . . 3  |-  ( rank `  A )  e.  On
14 limord 5444 . . . . . 6  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
152, 14ax-mp 5 . . . . 5  |-  Ord  dom  R1
16 ordelon 5409 . . . . 5  |-  ( ( Ord  dom  R1  /\  B  e.  dom  R1 )  ->  B  e.  On )
1715, 16mpan 674 . . . 4  |-  ( B  e.  dom  R1  ->  B  e.  On )
1817adantl 467 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  B  e.  On )
19 onsssuc 5472 . . 3  |-  ( ( ( rank `  A
)  e.  On  /\  B  e.  On )  ->  ( ( rank `  A
)  C_  B  <->  ( rank `  A )  e.  suc  B ) )
2013, 18, 19sylancr 667 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ( rank `  A )  C_  B  <->  (
rank `  A )  e.  suc  B ) )
216, 12, 203bitr4d 288 1  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  C_  ( R1 `  B )  <-> 
( rank `  A )  C_  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872    C_ wss 3379   ~Pcpw 3924   U.cuni 4162   dom cdm 4796   "cima 4799   Ord word 5384   Oncon0 5385   Lim wlim 5386   suc csuc 5387   Fun wfun 5538   ` cfv 5544   R1cr1 8185   rankcrnk 8186
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 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
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 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-om 6651  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-r1 8187  df-rank 8188
This theorem is referenced by:  r1rankidb  8227  rankval3b  8249  rankssb  8271  rankeq0b  8283  rankr1id  8285  rankr1b  8287
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