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Theorem rankf 8104
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 8075 . . . 4  |-  rank  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
21funmpt2 5555 . . 3  |-  Fun  rank
3 mptv 4484 . . . . . 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 2480 . . . . 5  |-  rank  =  { <. x ,  z
>.  |  z  =  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }
54dmeqi 5141 . . . 4  |-  dom  rank  =  dom  { <. x ,  z >.  |  z  =  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }
6 dmopab 5150 . . . . 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 2576 . . . . . 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 8103 . . . . . . 7  |-  ( x  e.  U. ( R1
" On )  <->  E. y  e.  On  x  e.  ( R1 `  suc  y
) )
9 intexrab 4551 . . . . . . 7  |-  ( E. y  e.  On  x  e.  ( R1 `  suc  y )  <->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  _V )
10 isset 3074 . . . . . . 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 272 . . . . . 6  |-  ( E. z  z  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  <-> 
x  e.  U. ( R1 " On ) )
127, 11mpgbir 1596 . . . . 5  |-  { x  |  E. z  z  = 
|^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }  =  U. ( R1 " On )
136, 12eqtri 2480 . . . 4  |-  dom  { <. x ,  z >.  |  z  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) } }  =  U. ( R1 " On )
145, 13eqtri 2480 . . 3  |-  dom  rank  = 
U. ( R1 " On )
15 df-fn 5521 . . 3  |-  ( rank 
Fn  U. ( R1 " On )  <->  ( Fun  rank  /\ 
dom  rank  =  U. ( R1 " On ) ) )
162, 14, 15mpbir2an 911 . 2  |-  rank  Fn  U. ( R1 " On )
17 rabn0 3757 . . . . 5  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  <->  E. y  e.  On  x  e.  ( R1 ` 
suc  y ) )
188, 17bitr4i 252 . . . 4  |-  ( x  e.  U. ( R1
" On )  <->  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =/=  (/) )
19 intex 4548 . . . . . 6  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  <->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  _V )
20 vex 3073 . . . . . . 7  |-  x  e. 
_V
211fvmpt2 5882 . . . . . . 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 670 . . . . . 6  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  e.  _V  ->  ( rank `  x )  = 
|^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
2319, 22sylbi 195 . . . . 5  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  ->  ( rank `  x )  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) } )
24 ssrab2 3537 . . . . . 6  |-  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  C_  On
25 oninton 6513 . . . . . 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 670 . . . . 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 195 . . 3  |-  ( x  e.  U. ( R1
" On )  -> 
( rank `  x )  e.  On )
2928rgen 2891 . 2  |-  A. x  e.  U. ( R1 " On ) ( rank `  x
)  e.  On
30 ffnfv 5970 . 2  |-  ( rank
: U. ( R1
" On ) --> On  <->  (
rank  Fn  U. ( R1 " On )  /\  A. x  e.  U. ( R1 " On ) (
rank `  x )  e.  On ) )
3116, 29, 30mpbir2an 911 1  |-  rank : U. ( R1 " On ) --> On
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2436    =/= wne 2644   A.wral 2795   E.wrex 2796   {crab 2799   _Vcvv 3070    C_ wss 3428   (/)c0 3737   U.cuni 4191   |^|cint 4228   {copab 4449    |-> cmpt 4450   Oncon0 4819   suc csuc 4821   dom cdm 4940   "cima 4943   Fun wfun 5512    Fn wfn 5513   -->wf 5514   ` cfv 5518   R1cr1 8072   rankcrnk 8073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-om 6579  df-recs 6934  df-rdg 6968  df-r1 8074  df-rank 8075
This theorem is referenced by:  rankon  8105  rankvaln  8109  tcrank  8194  hsmexlem4  8701  hsmexlem5  8702  grur1  9090  aomclem4  29550
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