Theorem rankwflemb 8656

Description: Two ways of saying a set is well-founded. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
rankwflemb	|- ( A e. U. ( R1 " On ) <-> E. x e. On A e. ( R1 ` suc x ) )

Distinct variable group:   x, A

Proof of Theorem rankwflemb

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 4439	. . 3 |- ( A e. U. ( R1 " On ) <-> E. y ( A e. y /\ y e. ( R1 " On ) ) )
2		r1funlim 8629	. . . . . . . 8 |- ( Fun R1 /\ Lim dom R1 )
3	2	simpli 474	. . . . . . 7 |- Fun R1
4		fvelima 6248	. . . . . . 7 |- ( ( Fun R1 /\ y e. ( R1 " On ) ) -> E. x e. On ( R1 ` x ) = y )
5	3, 4	mpan 706	. . . . . 6 |- ( y e. ( R1 " On ) -> E. x e. On ( R1 ` x ) = y )
6		eleq2 2690	. . . . . . . . 9 |- ( ( R1 ` x ) = y -> ( A e. ( R1 ` x ) <-> A e. y ) )
7	6	biimprcd 240	. . . . . . . 8 |- ( A e. y -> ( ( R1 ` x ) = y -> A e. ( R1 ` x ) ) )
8		r1tr 8639	. . . . . . . . . . . 12 |- Tr ( R1 ` x )
9		trss 4761	. . . . . . . . . . . 12 |- ( Tr ( R1 ` x ) -> ( A e. ( R1 ` x ) -> A C_ ( R1 ` x ) ) )
10	8, 9	ax-mp 5	. . . . . . . . . . 11 |- ( A e. ( R1 ` x ) -> A C_ ( R1 ` x ) )
11		elpwg 4166	. . . . . . . . . . 11 |- ( A e. ( R1 ` x ) -> ( A e. ~P ( R1 ` x ) <-> A C_ ( R1 ` x ) ) )
12	10, 11	mpbird 247	. . . . . . . . . 10 |- ( A e. ( R1 ` x ) -> A e. ~P ( R1 ` x ) )
13		elfvdm 6220	. . . . . . . . . . 11 |- ( A e. ( R1 ` x ) -> x e. dom R1 )
14		r1sucg 8632	. . . . . . . . . . 11 |- ( x e. dom R1 -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
15	13, 14	syl 17	. . . . . . . . . 10 |- ( A e. ( R1 ` x ) -> ( R1 ` suc x ) = ~P ( R1 ` x ) )
16	12, 15	eleqtrrd 2704	. . . . . . . . 9 |- ( A e. ( R1 ` x ) -> A e. ( R1 ` suc x ) )
17	16	a1i 11	. . . . . . . 8 |- ( x e. On -> ( A e. ( R1 ` x ) -> A e. ( R1 ` suc x ) ) )
18	7, 17	syl9 77	. . . . . . 7 |- ( A e. y -> ( x e. On -> ( ( R1 ` x ) = y -> A e. ( R1 ` suc x ) ) ) )
19	18	reximdvai 3015	. . . . . 6 |- ( A e. y -> ( E. x e. On ( R1 ` x ) = y -> E. x e. On A e. ( R1 ` suc x ) ) )
20	5, 19	syl5 34	. . . . 5 |- ( A e. y -> ( y e. ( R1 " On ) -> E. x e. On A e. ( R1 ` suc x ) ) )
21	20	imp 445	. . . 4 |- ( ( A e. y /\ y e. ( R1 " On ) ) -> E. x e. On A e. ( R1 ` suc x ) )
22	21	exlimiv 1858	. . 3 |- ( E. y ( A e. y /\ y e. ( R1 " On ) ) -> E. x e. On A e. ( R1 ` suc x ) )
23	1, 22	sylbi 207	. 2 |- ( A e. U. ( R1 " On ) -> E. x e. On A e. ( R1 ` suc x ) )
24		elfvdm 6220	. . . . . 6 |- ( A e. ( R1 ` suc x ) -> suc x e. dom R1 )
25		fvelrn 6352	. . . . . 6 |- ( ( Fun R1 /\ suc x e. dom R1 ) -> ( R1 ` suc x ) e. ran R1 )
26	3, 24, 25	sylancr 695	. . . . 5 |- ( A e. ( R1 ` suc x ) -> ( R1 ` suc x ) e. ran R1 )
27		df-ima 5127	. . . . . 6 |- ( R1 " On ) = ran ( R1 |` On )
28		funrel 5905	. . . . . . . . 9 |- ( Fun R1 -> Rel R1 )
29	3, 28	ax-mp 5	. . . . . . . 8 |- Rel R1
30	2	simpri 478	. . . . . . . . 9 |- Lim dom R1
31		limord 5784	. . . . . . . . 9 |- ( Lim dom R1 -> Ord dom R1 )
32		ordsson 6989	. . . . . . . . 9 |- ( Ord dom R1 -> dom R1 C_ On )
33	30, 31, 32	mp2b 10	. . . . . . . 8 |- dom R1 C_ On
34		relssres 5437	. . . . . . . 8 |- ( ( Rel R1 /\ dom R1 C_ On ) -> ( R1 |` On ) = R1 )
35	29, 33, 34	mp2an 708	. . . . . . 7 |- ( R1 |` On ) = R1
36	35	rneqi 5352	. . . . . 6 |- ran ( R1 |` On ) = ran R1
37	27, 36	eqtri 2644	. . . . 5 |- ( R1 " On ) = ran R1
38	26, 37	syl6eleqr 2712	. . . 4 |- ( A e. ( R1 ` suc x ) -> ( R1 ` suc x ) e. ( R1 " On ) )
39		elunii 4441	. . . 4 |- ( ( A e. ( R1 ` suc x ) /\ ( R1 ` suc x ) e. ( R1 " On ) ) -> A e. U. ( R1 " On ) )
40	38, 39	mpdan 702	. . 3 |- ( A e. ( R1 ` suc x ) -> A e. U. ( R1 " On ) )
41	40	rexlimivw 3029	. 2 |- ( E. x e. On A e. ( R1 ` suc x ) -> A e. U. ( R1 " On ) )
42	23, 41	impbii 199	1 |- ( A e. U. ( R1 " On ) <-> E. x e. On A e. ( R1 ` suc x ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   Tr wtr 4752   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Rel wrel 5119   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725   Fun wfun 5882   ` cfv 5888   R1cr1 8625

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-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

This theorem is referenced by:  rankf  8657  r1elwf  8659  rankvalb  8660  rankidb  8663  rankwflem  8678  tcrank  8747  dfac12r  8968