MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankr1ai Unicode version

Theorem rankr1ai 7470
Description: One direction of rankr1a 7508. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
rankr1ai  |-  ( A  e.  ( R1 `  B )  ->  ( rank `  A )  e.  B )

Proof of Theorem rankr1ai
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elfvdm 5554 . . 3  |-  ( A  e.  ( R1 `  B )  ->  B  e.  dom  R1 )
2 r1val1 7458 . . . . . 6  |-  ( B  e.  dom  R1  ->  ( R1 `  B )  =  U_ x  e.  B  ~P ( R1
`  x ) )
32eleq2d 2350 . . . . 5  |-  ( B  e.  dom  R1  ->  ( A  e.  ( R1
`  B )  <->  A  e.  U_ x  e.  B  ~P ( R1 `  x ) ) )
4 eliun 3909 . . . . 5  |-  ( A  e.  U_ x  e.  B  ~P ( R1
`  x )  <->  E. x  e.  B  A  e.  ~P ( R1 `  x
) )
53, 4syl6bb 252 . . . 4  |-  ( B  e.  dom  R1  ->  ( A  e.  ( R1
`  B )  <->  E. x  e.  B  A  e.  ~P ( R1 `  x
) ) )
6 r1funlim 7438 . . . . . . . . . . 11  |-  ( Fun 
R1  /\  Lim  dom  R1 )
76simpri 448 . . . . . . . . . 10  |-  Lim  dom  R1
8 limord 4451 . . . . . . . . . 10  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
97, 8ax-mp 8 . . . . . . . . 9  |-  Ord  dom  R1
10 ordtr1 4435 . . . . . . . . 9  |-  ( Ord 
dom  R1  ->  ( ( x  e.  B  /\  B  e.  dom  R1 )  ->  x  e.  dom  R1 ) )
119, 10ax-mp 8 . . . . . . . 8  |-  ( ( x  e.  B  /\  B  e.  dom  R1 )  ->  x  e.  dom  R1 )
1211ancoms 439 . . . . . . 7  |-  ( ( B  e.  dom  R1  /\  x  e.  B )  ->  x  e.  dom  R1 )
13 r1sucg 7441 . . . . . . . 8  |-  ( x  e.  dom  R1  ->  ( R1 `  suc  x
)  =  ~P ( R1 `  x ) )
1413eleq2d 2350 . . . . . . 7  |-  ( x  e.  dom  R1  ->  ( A  e.  ( R1
`  suc  x )  <->  A  e.  ~P ( R1
`  x ) ) )
1512, 14syl 15 . . . . . 6  |-  ( ( B  e.  dom  R1  /\  x  e.  B )  ->  ( A  e.  ( R1 `  suc  x )  <->  A  e.  ~P ( R1 `  x
) ) )
16 ordsson 4581 . . . . . . . . . 10  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
179, 16ax-mp 8 . . . . . . . . 9  |-  dom  R1  C_  On
1817, 12sseldi 3178 . . . . . . . 8  |-  ( ( B  e.  dom  R1  /\  x  e.  B )  ->  x  e.  On )
19 rabid 2716 . . . . . . . . 9  |-  ( x  e.  { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  <->  ( x  e.  On  /\  A  e.  ( R1 `  suc  x ) ) )
20 intss1 3877 . . . . . . . . 9  |-  ( x  e.  { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  ->  |^|
{ x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x )
2119, 20sylbir 204 . . . . . . . 8  |-  ( ( x  e.  On  /\  A  e.  ( R1 ` 
suc  x ) )  ->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x )
2218, 21sylan 457 . . . . . . 7  |-  ( ( ( B  e.  dom  R1 
/\  x  e.  B
)  /\  A  e.  ( R1 `  suc  x
) )  ->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x )
2322ex 423 . . . . . 6  |-  ( ( B  e.  dom  R1  /\  x  e.  B )  ->  ( A  e.  ( R1 `  suc  x )  ->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x ) )
2415, 23sylbird 226 . . . . 5  |-  ( ( B  e.  dom  R1  /\  x  e.  B )  ->  ( A  e. 
~P ( R1 `  x )  ->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x ) )
2524reximdva 2655 . . . 4  |-  ( B  e.  dom  R1  ->  ( E. x  e.  B  A  e.  ~P ( R1 `  x )  ->  E. x  e.  B  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x )
)
265, 25sylbid 206 . . 3  |-  ( B  e.  dom  R1  ->  ( A  e.  ( R1
`  B )  ->  E. x  e.  B  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x )
)
271, 26mpcom 32 . 2  |-  ( A  e.  ( R1 `  B )  ->  E. x  e.  B  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x )
28 r1elwf 7468 . . . . . . 7  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
29 rankvalb 7469 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
3028, 29syl 15 . . . . . 6  |-  ( A  e.  ( R1 `  B )  ->  ( rank `  A )  = 
|^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
3130sseq1d 3205 . . . . 5  |-  ( A  e.  ( R1 `  B )  ->  (
( rank `  A )  C_  x  <->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x ) )
3231adantr 451 . . . 4  |-  ( ( A  e.  ( R1
`  B )  /\  x  e.  B )  ->  ( ( rank `  A
)  C_  x  <->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x ) )
33 rankon 7467 . . . . . . 7  |-  ( rank `  A )  e.  On
3417, 1sseldi 3178 . . . . . . 7  |-  ( A  e.  ( R1 `  B )  ->  B  e.  On )
35 ontr2 4439 . . . . . . 7  |-  ( ( ( rank `  A
)  e.  On  /\  B  e.  On )  ->  ( ( ( rank `  A )  C_  x  /\  x  e.  B
)  ->  ( rank `  A )  e.  B
) )
3633, 34, 35sylancr 644 . . . . . 6  |-  ( A  e.  ( R1 `  B )  ->  (
( ( rank `  A
)  C_  x  /\  x  e.  B )  ->  ( rank `  A
)  e.  B ) )
3736exp3acom23 1362 . . . . 5  |-  ( A  e.  ( R1 `  B )  ->  (
x  e.  B  -> 
( ( rank `  A
)  C_  x  ->  (
rank `  A )  e.  B ) ) )
3837imp 418 . . . 4  |-  ( ( A  e.  ( R1
`  B )  /\  x  e.  B )  ->  ( ( rank `  A
)  C_  x  ->  (
rank `  A )  e.  B ) )
3932, 38sylbird 226 . . 3  |-  ( ( A  e.  ( R1
`  B )  /\  x  e.  B )  ->  ( |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x  ->  ( rank `  A
)  e.  B ) )
4039rexlimdva 2667 . 2  |-  ( A  e.  ( R1 `  B )  ->  ( E. x  e.  B  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x  ->  (
rank `  A )  e.  B ) )
4127, 40mpd 14 1  |-  ( A  e.  ( R1 `  B )  ->  ( rank `  A )  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   |^|cint 3862   U_ciun 3905   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:  rankr1ag  7474  tcrank  7554  dfac12lem1  7769  dfac12lem2  7770  r1limwun  8358  inatsk  8400  aomclem4  26566
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
  Copyright terms: Public domain W3C validator