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Theorem rankf 7482
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 7453 . . . 4  |-  rank  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
21funmpt2 5307 . . 3  |-  Fun  rank
3 mptv 4128 . . . . . 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 2316 . . . . 5  |-  rank  =  { <. x ,  z
>.  |  z  =  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }
54dmeqi 4896 . . . 4  |-  dom  rank  =  dom  { <. x ,  z >.  |  z  =  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }
6 dmopab 4905 . . . . 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 2402 . . . . . 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 7481 . . . . . . 7  |-  ( x  e.  U. ( R1
" On )  <->  E. y  e.  On  x  e.  ( R1 `  suc  y
) )
9 intexrab 4186 . . . . . . 7  |-  ( E. y  e.  On  x  e.  ( R1 `  suc  y )  <->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  _V )
10 isset 2805 . . . . . . 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 1540 . . . . 5  |-  { x  |  E. z  z  = 
|^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } }  =  U. ( R1 " On )
136, 12eqtri 2316 . . . 4  |-  dom  { <. x ,  z >.  |  z  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) } }  =  U. ( R1 " On )
145, 13eqtri 2316 . . 3  |-  dom  rank  = 
U. ( R1 " On )
15 df-fn 5274 . . 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 3487 . . . . 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 4183 . . . . . 6  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/)  <->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  _V )
20 vex 2804 . . . . . . 7  |-  x  e. 
_V
211fvmpt2 5624 . . . . . . 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 3271 . . . . . 6  |-  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  C_  On
25 oninton 4607 . . . . . 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 2370 . . . 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 2621 . 2  |-  A. x  e.  U. ( R1 " On ) ( rank `  x
)  e.  On
30 ffnfv 5701 . 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 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   U.cuni 3843   |^|cint 3878   {copab 4092    e. cmpt 4093   Oncon0 4408   suc csuc 4410   dom cdm 4705   "cima 4708   Fun wfun 5265    Fn wfn 5266   -->wf 5267   ` cfv 5271   R1cr1 7450   rankcrnk 7451
This theorem is referenced by:  rankon  7483  rankvaln  7487  tcrank  7570  hsmexlem4  8071  hsmexlem5  8072  grur1  8458  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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