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Theorem rankr1bg 7475
Description: A relationship between rank and  R1. See rankr1ag 7474 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 7438 . . . . 5  |-  ( Fun 
R1  /\  Lim  dom  R1 )
21simpri 448 . . . 4  |-  Lim  dom  R1
3 limsuc 4640 . . . 4  |-  ( Lim 
dom  R1  ->  ( B  e.  dom  R1  <->  suc  B  e. 
dom  R1 ) )
42, 3ax-mp 8 . . 3  |-  ( B  e.  dom  R1  <->  suc  B  e. 
dom  R1 )
5 rankr1ag 7474 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  suc  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  ( rank `  A
)  e.  suc  B
) )
64, 5sylan2b 461 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  ( rank `  A
)  e.  suc  B
) )
7 r1sucg 7441 . . . . 5  |-  ( B  e.  dom  R1  ->  ( R1 `  suc  B
)  =  ~P ( R1 `  B ) )
87adantl 452 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( R1 `  suc  B )  =  ~P ( R1 `  B ) )
98eleq2d 2350 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  e.  ( R1 `  suc  B )  <->  A  e.  ~P ( R1 `  B ) ) )
10 fvex 5539 . . . 4  |-  ( R1
`  B )  e. 
_V
1110elpw2 4175 . . 3  |-  ( A  e.  ~P ( R1
`  B )  <->  A  C_  ( R1 `  B ) )
129, 11syl6rbb 253 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  C_  ( R1 `  B )  <-> 
A  e.  ( R1
`  suc  B )
) )
13 rankon 7467 . . 3  |-  ( rank `  A )  e.  On
14 limord 4451 . . . . . 6  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
152, 14ax-mp 8 . . . . 5  |-  Ord  dom  R1
16 ordelon 4416 . . . . 5  |-  ( ( Ord  dom  R1  /\  B  e.  dom  R1 )  ->  B  e.  On )
1715, 16mpan 651 . . . 4  |-  ( B  e.  dom  R1  ->  B  e.  On )
1817adantl 452 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  B  e.  On )
19 onsssuc 4480 . . 3  |-  ( ( ( rank `  A
)  e.  On  /\  B  e.  On )  ->  ( ( rank `  A
)  C_  B  <->  ( rank `  A )  e.  suc  B ) )
2013, 18, 19sylancr 644 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( ( rank `  A )  C_  B  <->  (
rank `  A )  e.  suc  B ) )
216, 12, 203bitr4d 276 1  |-  ( ( A  e.  U. ( R1 " On )  /\  B  e.  dom  R1 )  ->  ( A  C_  ( R1 `  B )  <-> 
( rank `  A )  C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   Ord word 4391   Oncon0 4392   Lim wlim 4393   suc csuc 4394   dom cdm 4689   "cima 4692   Fun wfun 5249   ` cfv 5255   R1cr1 7434   rankcrnk 7435
This theorem is referenced by:  r1rankidb  7476  rankval3b  7498  rankssb  7520  rankeq0b  7532  rankr1id  7534  rankr1b  7536
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-int 3863  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  df-rank 7437
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