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Theorem rankf 7722
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 7693 . . . 4  |-  rank  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
21funmpt2 5492 . . 3  |-  Fun  rank
3 mptv 4303 . . . . . 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 2458 . . . . 5  |-  rank  =  { <. x ,  z
>.  |  z  =  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }
54dmeqi 5073 . . . 4  |-  dom  rank  =  dom  { <. x ,  z >.  |  z  =  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }
6 dmopab 5082 . . . . 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 2544 . . . . . 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 7721 . . . . . . 7  |-  ( x  e.  U. ( R1
" On )  <->  E. y  e.  On  x  e.  ( R1 `  suc  y
) )
9 intexrab 4361 . . . . . . 7  |-  ( E. y  e.  On  x  e.  ( R1 `  suc  y )  <->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  _V )
10 isset 2962 . . . . . . 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 265 . . . . . 6  |-  ( E. z  z  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  <-> 
x  e.  U. ( R1 " On ) )
127, 11mpgbir 1560 . . . . 5  |-  { x  |  E. z  z  = 
|^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }  =  U. ( R1 " On )
136, 12eqtri 2458 . . . 4  |-  dom  { <. x ,  z >.  |  z  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) } }  =  U. ( R1 " On )
145, 13eqtri 2458 . . 3  |-  dom  rank  = 
U. ( R1 " On )
15 df-fn 5459 . . 3  |-  ( rank 
Fn  U. ( R1 " On )  <->  ( Fun  rank  /\ 
dom  rank  =  U. ( R1 " On ) ) )
162, 14, 15mpbir2an 888 . 2  |-  rank  Fn  U. ( R1 " On )
17 rabn0 3649 . . . . 5  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  <->  E. y  e.  On  x  e.  ( R1 ` 
suc  y ) )
188, 17bitr4i 245 . . . 4  |-  ( x  e.  U. ( R1
" On )  <->  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =/=  (/) )
19 intex 4358 . . . . . 6  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  <->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  _V )
20 vex 2961 . . . . . . 7  |-  x  e. 
_V
211fvmpt2 5814 . . . . . . 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 653 . . . . . 6  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  e.  _V  ->  ( rank `  x )  = 
|^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
2319, 22sylbi 189 . . . . 5  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  ->  ( rank `  x )  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) } )
24 ssrab2 3430 . . . . . 6  |-  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  C_  On
25 oninton 4782 . . . . . 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 653 . . . . 5  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  On )
2723, 26eqeltrd 2512 . . . 4  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  ->  ( rank `  x )  e.  On )
2818, 27sylbi 189 . . 3  |-  ( x  e.  U. ( R1
" On )  -> 
( rank `  x )  e.  On )
2928rgen 2773 . 2  |-  A. x  e.  U. ( R1 " On ) ( rank `  x
)  e.  On
30 ffnfv 5896 . 2  |-  ( rank
: U. ( R1
" On ) --> On  <->  (
rank  Fn  U. ( R1 " On )  /\  A. x  e.  U. ( R1 " On ) (
rank `  x )  e.  On ) )
3116, 29, 30mpbir2an 888 1  |-  rank : U. ( R1 " On ) --> On
Colors of variables: wff set class
Syntax hints:    <-> wb 178   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711   _Vcvv 2958    C_ wss 3322   (/)c0 3630   U.cuni 4017   |^|cint 4052   {copab 4267    e. cmpt 4268   Oncon0 4583   suc csuc 4585   dom cdm 4880   "cima 4883   Fun wfun 5450    Fn wfn 5451   -->wf 5452   ` cfv 5456   R1cr1 7690   rankcrnk 7691
This theorem is referenced by:  rankon  7723  rankvaln  7727  tcrank  7810  hsmexlem4  8311  hsmexlem5  8312  grur1  8697  aomclem4  27134
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-recs 6635  df-rdg 6670  df-r1 7692  df-rank 7693
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