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Theorem rankf 7466
Description: The domain and range of the  rank function. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 12-Sep-2013.)
Assertion
Ref Expression
rankf  |-  rank : U. ( R1 " On ) --> On

Proof of Theorem rankf
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rank 7437 . . . 4  |-  rank  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
21funmpt2 5291 . . 3  |-  Fun  rank
3 mptv 4112 . . . . . 6  |-  ( x  e.  _V  |->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } )  =  { <. x ,  z >.  |  z  =  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }
41, 3eqtri 2303 . . . . 5  |-  rank  =  { <. x ,  z
>.  |  z  =  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }
54dmeqi 4880 . . . 4  |-  dom  rank  =  dom  { <. x ,  z >.  |  z  =  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }
6 dmopab 4889 . . . . 5  |-  dom  { <. x ,  z >.  |  z  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) } }  =  { x  |  E. z  z  = 
|^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }
7 abeq1 2389 . . . . . 6  |-  ( { x  |  E. z 
z  =  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }  =  U. ( R1 " On )  <->  A. x ( E. z  z  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  <-> 
x  e.  U. ( R1 " On ) ) )
8 rankwflemb 7465 . . . . . . 7  |-  ( x  e.  U. ( R1
" On )  <->  E. y  e.  On  x  e.  ( R1 `  suc  y
) )
9 intexrab 4170 . . . . . . 7  |-  ( E. y  e.  On  x  e.  ( R1 `  suc  y )  <->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  _V )
10 isset 2792 . . . . . . 7  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  e.  _V  <->  E. z 
z  =  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
118, 9, 103bitrri 263 . . . . . 6  |-  ( E. z  z  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  <-> 
x  e.  U. ( R1 " On ) )
127, 11mpgbir 1537 . . . . 5  |-  { x  |  E. z  z  = 
|^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }  =  U. ( R1 " On )
136, 12eqtri 2303 . . . 4  |-  dom  { <. x ,  z >.  |  z  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) } }  =  U. ( R1 " On )
145, 13eqtri 2303 . . 3  |-  dom  rank  = 
U. ( R1 " On )
15 df-fn 5258 . . 3  |-  ( rank 
Fn  U. ( R1 " On )  <->  ( Fun  rank  /\ 
dom  rank  =  U. ( R1 " On ) ) )
162, 14, 15mpbir2an 886 . 2  |-  rank  Fn  U. ( R1 " On )
17 rabn0 3474 . . . . 5  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  <->  E. y  e.  On  x  e.  ( R1 ` 
suc  y ) )
188, 17bitr4i 243 . . . 4  |-  ( x  e.  U. ( R1
" On )  <->  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =/=  (/) )
19 intex 4167 . . . . . 6  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  <->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  _V )
20 vex 2791 . . . . . . 7  |-  x  e. 
_V
211fvmpt2 5608 . . . . . . 7  |-  ( ( x  e.  _V  /\  |^|
{ y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  _V )  ->  ( rank `  x
)  =  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
2220, 21mpan 651 . . . . . 6  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  e.  _V  ->  ( rank `  x )  = 
|^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
2319, 22sylbi 187 . . . . 5  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  ->  ( rank `  x )  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) } )
24 ssrab2 3258 . . . . . 6  |-  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  C_  On
25 oninton 4591 . . . . . 6  |-  ( ( { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  C_  On  /\  {
y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/) )  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  On )
2624, 25mpan 651 . . . . 5  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  On )
2723, 26eqeltrd 2357 . . . 4  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  ->  ( rank `  x )  e.  On )
2818, 27sylbi 187 . . 3  |-  ( x  e.  U. ( R1
" On )  -> 
( rank `  x )  e.  On )
2928rgen 2608 . 2  |-  A. x  e.  U. ( R1 " On ) ( rank `  x
)  e.  On
30 ffnfv 5685 . 2  |-  ( rank
: U. ( R1
" On ) --> On  <->  (
rank  Fn  U. ( R1 " On )  /\  A. x  e.  U. ( R1 " On ) (
rank `  x )  e.  On ) )
3116, 29, 30mpbir2an 886 1  |-  rank : U. ( R1 " On ) --> On
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   U.cuni 3827   |^|cint 3862   {copab 4076    e. cmpt 4077   Oncon0 4392   suc csuc 4394   dom cdm 4689   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255   R1cr1 7434   rankcrnk 7435
This theorem is referenced by:  rankon  7467  rankvaln  7471  tcrank  7554  hsmexlem4  8055  hsmexlem5  8056  grur1  8442  aomclem4  27154
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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