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Theorem rankf 8290
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 8261 . . . 4  |-  rank  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
21funmpt2 5637 . . 3  |-  Fun  rank
3 mptv 4509 . . . . . 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 2483 . . . . 5  |-  rank  =  { <. x ,  z
>.  |  z  =  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }
54dmeqi 5054 . . . 4  |-  dom  rank  =  dom  { <. x ,  z >.  |  z  =  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }
6 dmopab 5063 . . . . 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 2571 . . . . . 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 8289 . . . . . . 7  |-  ( x  e.  U. ( R1
" On )  <->  E. y  e.  On  x  e.  ( R1 `  suc  y
) )
9 intexrab 4575 . . . . . . 7  |-  ( E. y  e.  On  x  e.  ( R1 `  suc  y )  <->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  _V )
10 isset 3060 . . . . . . 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 280 . . . . . 6  |-  ( E. z  z  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  <-> 
x  e.  U. ( R1 " On ) )
127, 11mpgbir 1683 . . . . 5  |-  { x  |  E. z  z  = 
|^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }  =  U. ( R1 " On )
136, 12eqtri 2483 . . . 4  |-  dom  { <. x ,  z >.  |  z  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) } }  =  U. ( R1 " On )
145, 13eqtri 2483 . . 3  |-  dom  rank  = 
U. ( R1 " On )
15 df-fn 5603 . . 3  |-  ( rank 
Fn  U. ( R1 " On )  <->  ( Fun  rank  /\ 
dom  rank  =  U. ( R1 " On ) ) )
162, 14, 15mpbir2an 936 . 2  |-  rank  Fn  U. ( R1 " On )
17 rabn0 3763 . . . . 5  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  <->  E. y  e.  On  x  e.  ( R1 ` 
suc  y ) )
188, 17bitr4i 260 . . . 4  |-  ( x  e.  U. ( R1
" On )  <->  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =/=  (/) )
19 intex 4572 . . . . . 6  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  <->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  _V )
20 vex 3059 . . . . . . 7  |-  x  e. 
_V
211fvmpt2 5979 . . . . . . 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 681 . . . . . 6  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  e.  _V  ->  ( rank `  x )  = 
|^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
2319, 22sylbi 200 . . . . 5  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  ->  ( rank `  x )  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) } )
24 ssrab2 3525 . . . . . 6  |-  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  C_  On
25 oninton 6653 . . . . . 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 681 . . . . 5  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  On )
2723, 26eqeltrd 2539 . . . 4  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  ->  ( rank `  x )  e.  On )
2818, 27sylbi 200 . . 3  |-  ( x  e.  U. ( R1
" On )  -> 
( rank `  x )  e.  On )
2928rgen 2758 . 2  |-  A. x  e.  U. ( R1 " On ) ( rank `  x
)  e.  On
30 ffnfv 6071 . 2  |-  ( rank
: U. ( R1
" On ) --> On  <->  (
rank  Fn  U. ( R1 " On )  /\  A. x  e.  U. ( R1 " On ) (
rank `  x )  e.  On ) )
3116, 29, 30mpbir2an 936 1  |-  rank : U. ( R1 " On ) --> On
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    = wceq 1454   E.wex 1673    e. wcel 1897   {cab 2447    =/= wne 2632   A.wral 2748   E.wrex 2749   {crab 2752   _Vcvv 3056    C_ wss 3415   (/)c0 3742   U.cuni 4211   |^|cint 4247   {copab 4473    |-> cmpt 4474   dom cdm 4852   "cima 4855   Oncon0 5441   suc csuc 5443   Fun wfun 5594    Fn wfn 5595   -->wf 5596   ` cfv 5600   R1cr1 8258   rankcrnk 8259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-om 6719  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-r1 8260  df-rank 8261
This theorem is referenced by:  rankon  8291  rankvaln  8295  tcrank  8380  hsmexlem4  8884  hsmexlem5  8885  grur1  9270  aomclem4  35959
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