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

Theorem rankr1ai 7716
Description: One direction of rankr1a 7754. (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 5749 . . 3  |-  ( A  e.  ( R1 `  B )  ->  B  e.  dom  R1 )
2 r1val1 7704 . . . . . 6  |-  ( B  e.  dom  R1  ->  ( R1 `  B )  =  U_ x  e.  B  ~P ( R1
`  x ) )
32eleq2d 2502 . . . . 5  |-  ( B  e.  dom  R1  ->  ( A  e.  ( R1
`  B )  <->  A  e.  U_ x  e.  B  ~P ( R1 `  x ) ) )
4 eliun 4089 . . . . 5  |-  ( A  e.  U_ x  e.  B  ~P ( R1
`  x )  <->  E. x  e.  B  A  e.  ~P ( R1 `  x
) )
53, 4syl6bb 253 . . . 4  |-  ( B  e.  dom  R1  ->  ( A  e.  ( R1
`  B )  <->  E. x  e.  B  A  e.  ~P ( R1 `  x
) ) )
6 r1funlim 7684 . . . . . . . . . . 11  |-  ( Fun 
R1  /\  Lim  dom  R1 )
76simpri 449 . . . . . . . . . 10  |-  Lim  dom  R1
8 limord 4632 . . . . . . . . . 10  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
97, 8ax-mp 8 . . . . . . . . 9  |-  Ord  dom  R1
10 ordtr1 4616 . . . . . . . . 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 440 . . . . . . 7  |-  ( ( B  e.  dom  R1  /\  x  e.  B )  ->  x  e.  dom  R1 )
13 r1sucg 7687 . . . . . . . 8  |-  ( x  e.  dom  R1  ->  ( R1 `  suc  x
)  =  ~P ( R1 `  x ) )
1413eleq2d 2502 . . . . . . 7  |-  ( x  e.  dom  R1  ->  ( A  e.  ( R1
`  suc  x )  <->  A  e.  ~P ( R1
`  x ) ) )
1512, 14syl 16 . . . . . 6  |-  ( ( B  e.  dom  R1  /\  x  e.  B )  ->  ( A  e.  ( R1 `  suc  x )  <->  A  e.  ~P ( R1 `  x
) ) )
16 ordsson 4762 . . . . . . . . . 10  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
179, 16ax-mp 8 . . . . . . . . 9  |-  dom  R1  C_  On
1817, 12sseldi 3338 . . . . . . . 8  |-  ( ( B  e.  dom  R1  /\  x  e.  B )  ->  x  e.  On )
19 rabid 2876 . . . . . . . . 9  |-  ( x  e.  { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  <->  ( x  e.  On  /\  A  e.  ( R1 `  suc  x ) ) )
20 intss1 4057 . . . . . . . . 9  |-  ( x  e.  { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  ->  |^|
{ x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x )
2119, 20sylbir 205 . . . . . . . 8  |-  ( ( x  e.  On  /\  A  e.  ( R1 ` 
suc  x ) )  ->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x )
2218, 21sylan 458 . . . . . . 7  |-  ( ( ( B  e.  dom  R1 
/\  x  e.  B
)  /\  A  e.  ( R1 `  suc  x
) )  ->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x )
2322ex 424 . . . . . 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 227 . . . . 5  |-  ( ( B  e.  dom  R1  /\  x  e.  B )  ->  ( A  e. 
~P ( R1 `  x )  ->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x ) )
2524reximdva 2810 . . . 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 207 . . 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 34 . 2  |-  ( A  e.  ( R1 `  B )  ->  E. x  e.  B  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x )
28 r1elwf 7714 . . . . . . 7  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
29 rankvalb 7715 . . . . . . 7  |-  ( A  e.  U. ( R1
" On )  -> 
( rank `  A )  =  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
3028, 29syl 16 . . . . . 6  |-  ( A  e.  ( R1 `  B )  ->  ( rank `  A )  = 
|^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) } )
3130sseq1d 3367 . . . . 5  |-  ( A  e.  ( R1 `  B )  ->  (
( rank `  A )  C_  x  <->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x ) )
3231adantr 452 . . . 4  |-  ( ( A  e.  ( R1
`  B )  /\  x  e.  B )  ->  ( ( rank `  A
)  C_  x  <->  |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x ) )
33 rankon 7713 . . . . . . 7  |-  ( rank `  A )  e.  On
3417, 1sseldi 3338 . . . . . . 7  |-  ( A  e.  ( R1 `  B )  ->  B  e.  On )
35 ontr2 4620 . . . . . . 7  |-  ( ( ( rank `  A
)  e.  On  /\  B  e.  On )  ->  ( ( ( rank `  A )  C_  x  /\  x  e.  B
)  ->  ( rank `  A )  e.  B
) )
3633, 34, 35sylancr 645 . . . . . 6  |-  ( A  e.  ( R1 `  B )  ->  (
( ( rank `  A
)  C_  x  /\  x  e.  B )  ->  ( rank `  A
)  e.  B ) )
3736exp3acom23 1381 . . . . 5  |-  ( A  e.  ( R1 `  B )  ->  (
x  e.  B  -> 
( ( rank `  A
)  C_  x  ->  (
rank `  A )  e.  B ) ) )
3837imp 419 . . . 4  |-  ( ( A  e.  ( R1
`  B )  /\  x  e.  B )  ->  ( ( rank `  A
)  C_  x  ->  (
rank `  A )  e.  B ) )
3932, 38sylbird 227 . . 3  |-  ( ( A  e.  ( R1
`  B )  /\  x  e.  B )  ->  ( |^| { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  C_  x  ->  ( rank `  A
)  e.  B ) )
4039rexlimdva 2822 . 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 15 1  |-  ( A  e.  ( R1 `  B )  ->  ( rank `  A )  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   {crab 2701    C_ wss 3312   ~Pcpw 3791   U.cuni 4007   |^|cint 4042   U_ciun 4085   Ord word 4572   Oncon0 4573   Lim wlim 4574   suc csuc 4575   dom cdm 4870   "cima 4873   Fun wfun 5440   ` cfv 5446   R1cr1 7680   rankcrnk 7681
This theorem is referenced by:  rankr1ag  7720  tcrank  7800  dfac12lem1  8015  dfac12lem2  8016  r1limwun  8603  inatsk  8645  aomclem4  27123
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-int 4043  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  df-rank 7683
  Copyright terms: Public domain W3C validator