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Theorem rankr1ai 7486
Description: One direction of rankr1a 7524. (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 5570 . . 3  |-  ( A  e.  ( R1 `  B )  ->  B  e.  dom  R1 )
2 r1val1 7474 . . . . . 6  |-  ( B  e.  dom  R1  ->  ( R1 `  B )  =  U_ x  e.  B  ~P ( R1
`  x ) )
32eleq2d 2363 . . . . 5  |-  ( B  e.  dom  R1  ->  ( A  e.  ( R1
`  B )  <->  A  e.  U_ x  e.  B  ~P ( R1 `  x ) ) )
4 eliun 3925 . . . . 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 7454 . . . . . . . . . . 11  |-  ( Fun 
R1  /\  Lim  dom  R1 )
76simpri 448 . . . . . . . . . 10  |-  Lim  dom  R1
8 limord 4467 . . . . . . . . . 10  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
97, 8ax-mp 8 . . . . . . . . 9  |-  Ord  dom  R1
10 ordtr1 4451 . . . . . . . . 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 7457 . . . . . . . 8  |-  ( x  e.  dom  R1  ->  ( R1 `  suc  x
)  =  ~P ( R1 `  x ) )
1413eleq2d 2363 . . . . . . 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 4597 . . . . . . . . . 10  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
179, 16ax-mp 8 . . . . . . . . 9  |-  dom  R1  C_  On
1817, 12sseldi 3191 . . . . . . . 8  |-  ( ( B  e.  dom  R1  /\  x  e.  B )  ->  x  e.  On )
19 rabid 2729 . . . . . . . . 9  |-  ( x  e.  { x  e.  On  |  A  e.  ( R1 `  suc  x ) }  <->  ( x  e.  On  /\  A  e.  ( R1 `  suc  x ) ) )
20 intss1 3893 . . . . . . . . 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 2668 . . . 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 7484 . . . . . . 7  |-  ( A  e.  ( R1 `  B )  ->  A  e.  U. ( R1 " On ) )
29 rankvalb 7485 . . . . . . 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 3218 . . . . 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 7483 . . . . . . 7  |-  ( rank `  A )  e.  On
3417, 1sseldi 3191 . . . . . . 7  |-  ( A  e.  ( R1 `  B )  ->  B  e.  On )
35 ontr2 4455 . . . . . . 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 2680 . 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 1632    e. wcel 1696   E.wrex 2557   {crab 2560    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   |^|cint 3878   U_ciun 3921   Ord word 4407   Oncon0 4408   Lim wlim 4409   suc csuc 4410   dom cdm 4705   "cima 4708   Fun wfun 5265   ` cfv 5271   R1cr1 7450   rankcrnk 7451
This theorem is referenced by:  rankr1ag  7490  tcrank  7570  dfac12lem1  7785  dfac12lem2  7786  r1limwun  8374  inatsk  8416  aomclem4  27257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439  df-r1 7452  df-rank 7453
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