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Theorem rankr1ai 7658
Description: One direction of rankr1a 7696. (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 5698 . . 3  |-  ( A  e.  ( R1 `  B )  ->  B  e.  dom  R1 )
2 r1val1 7646 . . . . . 6  |-  ( B  e.  dom  R1  ->  ( R1 `  B )  =  U_ x  e.  B  ~P ( R1
`  x ) )
32eleq2d 2455 . . . . 5  |-  ( B  e.  dom  R1  ->  ( A  e.  ( R1
`  B )  <->  A  e.  U_ x  e.  B  ~P ( R1 `  x ) ) )
4 eliun 4040 . . . . 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 7626 . . . . . . . . . . 11  |-  ( Fun 
R1  /\  Lim  dom  R1 )
76simpri 449 . . . . . . . . . 10  |-  Lim  dom  R1
8 limord 4582 . . . . . . . . . 10  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
97, 8ax-mp 8 . . . . . . . . 9  |-  Ord  dom  R1
10 ordtr1 4566 . . . . . . . . 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 7629 . . . . . . . 8  |-  ( x  e.  dom  R1  ->  ( R1 `  suc  x
)  =  ~P ( R1 `  x ) )
1413eleq2d 2455 . . . . . . 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 4711 . . . . . . . . . 10  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
179, 16ax-mp 8 . . . . . . . . 9  |-  dom  R1  C_  On
1817, 12sseldi 3290 . . . . . . . 8  |-  ( ( B  e.  dom  R1  /\  x  e.  B )  ->  x  e.  On )
19 rabid 2828 . . . . . . . . 9  |-  ( x  e.  { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  <->  ( x  e.  On  /\  A  e.  ( R1 `  suc  x ) ) )
20 intss1 4008 . . . . . . . . 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 2762 . . . 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 7656 . . . . . . 7  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
29 rankvalb 7657 . . . . . . 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 3319 . . . . 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 7655 . . . . . . 7  |-  ( rank `  A )  e.  On
3417, 1sseldi 3290 . . . . . . 7  |-  ( A  e.  ( R1 `  B )  ->  B  e.  On )
35 ontr2 4570 . . . . . . 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 1378 . . . . 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 2774 . 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 1649    e. wcel 1717   E.wrex 2651   {crab 2654    C_ wss 3264   ~Pcpw 3743   U.cuni 3958   |^|cint 3993   U_ciun 4036   Ord word 4522   Oncon0 4523   Lim wlim 4524   suc csuc 4525   dom cdm 4819   "cima 4822   Fun wfun 5389   ` cfv 5395   R1cr1 7622   rankcrnk 7623
This theorem is referenced by:  rankr1ag  7662  tcrank  7742  dfac12lem1  7957  dfac12lem2  7958  r1limwun  8545  inatsk  8587  aomclem4  26824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-recs 6570  df-rdg 6605  df-r1 7624  df-rank 7625
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