Theorem rankf 8657

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.

Step	Hyp	Ref	Expression
1		df-rank 8628	. . . 4 |- rank = ( x e. _V |-> |^| { y e. On | x e. ( R1 ` suc y ) } )
2	1	funmpt2 5927	. . 3 |- Fun rank
3		mptv 4751	. . . . . 6 |- ( x e. _V |-> |^| { y e. On | x e. ( R1 ` suc y ) } ) = { <. x , z >. | z = |^| { y e. On | x e. ( R1 ` suc y ) } }
4	1, 3	eqtri 2644	. . . . 5 |- rank = { <. x , z >. | z = |^| { y e. On | x e. ( R1 ` suc y ) } }
5	4	dmeqi 5325	. . . 4 |- dom rank = dom { <. x , z >. | z = |^| { y e. On | x e. ( R1 ` suc y ) } }
6		dmopab 5335	. . . . 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 2733	. . . . . 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 8656	. . . . . . 7 |- ( x e. U. ( R1 " On ) <-> E. y e. On x e. ( R1 ` suc y ) )
9		intexrab 4823	. . . . . . 7 |- ( E. y e. On x e. ( R1 ` suc y ) <-> |^| { y e. On | x e. ( R1 ` suc y ) } e. _V )
10		isset 3207	. . . . . . 7 |- ( |^| { y e. On | x e. ( R1 ` suc y ) } e. _V <-> E. z z = |^| { y e. On | x e. ( R1 ` suc y ) } )
11	8, 9, 10	3bitrri 287	. . . . . 6 |- ( E. z z = |^| { y e. On | x e. ( R1 ` suc y ) } <-> x e. U. ( R1 " On ) )
12	7, 11	mpgbir 1726	. . . . 5 |- { x | E. z z = |^| { y e. On | x e. ( R1 ` suc y ) } } = U. ( R1 " On )
13	6, 12	eqtri 2644	. . . 4 |- dom { <. x , z >. | z = |^| { y e. On | x e. ( R1 ` suc y ) } } = U. ( R1 " On )
14	5, 13	eqtri 2644	. . 3 |- dom rank = U. ( R1 " On )
15		df-fn 5891	. . 3 |- ( rank Fn U. ( R1 " On ) <-> ( Fun rank /\ dom rank = U. ( R1 " On ) ) )
16	2, 14, 15	mpbir2an 955	. 2 |- rank Fn U. ( R1 " On )
17		rabn0 3958	. . . . 5 |- ( { y e. On | x e. ( R1 ` suc y ) } =/= (/) <-> E. y e. On x e. ( R1 ` suc y ) )
18	8, 17	bitr4i 267	. . . 4 |- ( x e. U. ( R1 " On ) <-> { y e. On | x e. ( R1 ` suc y ) } =/= (/) )
19		intex 4820	. . . . . 6 |- ( { y e. On | x e. ( R1 ` suc y ) } =/= (/) <-> |^| { y e. On | x e. ( R1 ` suc y ) } e. _V )
20		vex 3203	. . . . . . 7 |- x e. _V
21	1	fvmpt2 6291	. . . . . . 7 |- ( ( x e. _V /\ |^| { y e. On | x e. ( R1 ` suc y ) } e. _V ) -> ( rank ` x ) = |^| { y e. On | x e. ( R1 ` suc y ) } )
22	20, 21	mpan 706	. . . . . 6 |- ( |^| { y e. On | x e. ( R1 ` suc y ) } e. _V -> ( rank ` x ) = |^| { y e. On | x e. ( R1 ` suc y ) } )
23	19, 22	sylbi 207	. . . . 5 |- ( { y e. On | x e. ( R1 ` suc y ) } =/= (/) -> ( rank ` x ) = |^| { y e. On | x e. ( R1 ` suc y ) } )
24		ssrab2 3687	. . . . . 6 |- { y e. On | x e. ( R1 ` suc y ) } C_ On
25		oninton 7000	. . . . . 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 )
26	24, 25	mpan 706	. . . . 5 |- ( { y e. On | x e. ( R1 ` suc y ) } =/= (/) -> |^| { y e. On | x e. ( R1 ` suc y ) } e. On )
27	23, 26	eqeltrd 2701	. . . 4 |- ( { y e. On | x e. ( R1 ` suc y ) } =/= (/) -> ( rank ` x ) e. On )
28	18, 27	sylbi 207	. . 3 |- ( x e. U. ( R1 " On ) -> ( rank ` x ) e. On )
29	28	rgen 2922	. 2 |- A. x e. U. ( R1 " On ) ( rank ` x ) e. On
30		ffnfv 6388	. 2 |- ( rank : U. ( R1 " On ) --> On <-> ( rank Fn U. ( R1 " On ) /\ A. x e. U. ( R1 " On ) ( rank ` x ) e. On ) )
31	16, 29, 30	mpbir2an 955	1 |- rank : U. ( R1 " On ) --> On

Syntax hints:   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   U.cuni 4436   |^|cint 4475   {copab 4712   |-> cmpt 4729   dom cdm 5114   "cima 5117   Oncon0 5723   suc csuc 5725   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888   R1cr1 8625   rankcrnk 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-r1 8627  df-rank 8628

This theorem is referenced by:  rankon  8658  rankvaln  8662  tcrank  8747  hsmexlem4  9251  hsmexlem5  9252  grur1  9642  aomclem4  37627