Theorem rankvalb 8293

Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 8312 does not use Regularity, and so requires the assumption that A is in the range of R1. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)

Assertion

Ref	Expression
rankvalb	|- ( A e. U. ( R1 " On ) -> ( rank ` A ) = |^| { x e. On | A e. ( R1 ` suc x ) } )

Distinct variable group:   x, A

Proof of Theorem rankvalb

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3065	. 2 |- ( A e. U. ( R1 " On ) -> A e. _V )
2		rankwflemb 8289	. . 3 |- ( A e. U. ( R1 " On ) <-> E. x e. On A e. ( R1 ` suc x ) )
3		intexrab 4575	. . 3 |- ( E. x e. On A e. ( R1 ` suc x ) <-> |^| { x e. On | A e. ( R1 ` suc x ) } e. _V )
4	2, 3	sylbb 202	. 2 |- ( A e. U. ( R1 " On ) -> |^| { x e. On | A e. ( R1 ` suc x ) } e. _V )
5		eleq1 2527	. . . . 5 |- ( y = A -> ( y e. ( R1 ` suc x ) <-> A e. ( R1 ` suc x ) ) )
6	5	rabbidv 3047	. . . 4 |- ( y = A -> { x e. On | y e. ( R1 ` suc x ) } = { x e. On | A e. ( R1 ` suc x ) } )
7	6	inteqd 4252	. . 3 |- ( y = A -> |^| { x e. On | y e. ( R1 ` suc x ) } = |^| { x e. On | A e. ( R1 ` suc x ) } )
8		df-rank 8261	. . 3 |- rank = ( y e. _V |-> |^| { x e. On | y e. ( R1 ` suc x ) } )
9	7, 8	fvmptg 5968	. 2 |- ( ( A e. _V /\ |^| { x e. On | A e. ( R1 ` suc x ) } e. _V ) -> ( rank ` A ) = |^| { x e. On | A e. ( R1 ` suc x ) } )
10	1, 4, 9	syl2anc 671	1 |- ( A e. U. ( R1 " On ) -> ( rank ` A ) = |^| { x e. On | A e. ( R1 ` suc x ) } )

Syntax hints:   -> wi 4   = wceq 1454   e. wcel 1897   E.wrex 2749   {crab 2752   _Vcvv 3056   U.cuni 4211   |^|cint 4247   "cima 4855   Oncon0 5441   suc csuc 5443   ` 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:  rankr1ai  8294  rankidb  8296  rankval  8312