Theorem wfrlem10 7424

Description: Lemma for well-founded recursion. When z is an R minimal element of ( A \ dom F ), then its predecessor class is equal to dom F. (Contributed by Scott Fenton, 21-Apr-2011.)

Hypotheses

Ref	Expression
wfrlem10.1	|- R We A
wfrlem10.2	|- F = wrecs ( R , A , G )

Assertion

Ref	Expression
wfrlem10	|- ( ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) -> Pred ( R , A , z ) = dom F )

Distinct variable group:   z, A

Allowed substitution hints:   R( z)   F( z)   G( z)

Proof of Theorem wfrlem10

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wfrlem10.2	. . . 4 |- F = wrecs ( R , A , G )
2	1	wfrlem8 7422	. . 3 |- ( Pred ( R , ( A \ dom F ) , z ) = (/) <-> Pred ( R , A , z ) = Pred ( R , dom F , z ) )
3	2	biimpi 206	. 2 |- ( Pred ( R , ( A \ dom F ) , z ) = (/) -> Pred ( R , A , z ) = Pred ( R , dom F , z ) )
4		predss 5687	. . . 4 |- Pred ( R , dom F , z ) C_ dom F
5	4	a1i 11	. . 3 |- ( z e. ( A \ dom F ) -> Pred ( R , dom F , z ) C_ dom F )
6		simpr 477	. . . . . 6 |- ( ( z e. ( A \ dom F ) /\ w e. dom F ) -> w e. dom F )
7		eldifn 3733	. . . . . . . . . 10 |- ( z e. ( A \ dom F ) -> -. z e. dom F )
8		eleq1 2689	. . . . . . . . . . 11 |- ( w = z -> ( w e. dom F <-> z e. dom F ) )
9	8	notbid 308	. . . . . . . . . 10 |- ( w = z -> ( -. w e. dom F <-> -. z e. dom F ) )
10	7, 9	syl5ibrcom 237	. . . . . . . . 9 |- ( z e. ( A \ dom F ) -> ( w = z -> -. w e. dom F ) )
11	10	con2d 129	. . . . . . . 8 |- ( z e. ( A \ dom F ) -> ( w e. dom F -> -. w = z ) )
12	11	imp 445	. . . . . . 7 |- ( ( z e. ( A \ dom F ) /\ w e. dom F ) -> -. w = z )
13	1	wfrdmcl 7423	. . . . . . . . . 10 |- ( w e. dom F -> Pred ( R , A , w ) C_ dom F )
14	13	adantl 482	. . . . . . . . 9 |- ( ( z e. ( A \ dom F ) /\ w e. dom F ) -> Pred ( R , A , w ) C_ dom F )
15		ssel 3597	. . . . . . . . . . . 12 |- ( Pred ( R , A , w ) C_ dom F -> ( z e. Pred ( R , A , w ) -> z e. dom F ) )
16	15	con3d 148	. . . . . . . . . . 11 |- ( Pred ( R , A , w ) C_ dom F -> ( -. z e. dom F -> -. z e. Pred ( R , A , w ) ) )
17	7, 16	syl5com 31	. . . . . . . . . 10 |- ( z e. ( A \ dom F ) -> ( Pred ( R , A , w ) C_ dom F -> -. z e. Pred ( R , A , w ) ) )
18	17	adantr 481	. . . . . . . . 9 |- ( ( z e. ( A \ dom F ) /\ w e. dom F ) -> ( Pred ( R , A , w ) C_ dom F -> -. z e. Pred ( R , A , w ) ) )
19	14, 18	mpd 15	. . . . . . . 8 |- ( ( z e. ( A \ dom F ) /\ w e. dom F ) -> -. z e. Pred ( R , A , w ) )
20		eldifi 3732	. . . . . . . . 9 |- ( z e. ( A \ dom F ) -> z e. A )
21		elpredg 5694	. . . . . . . . . 10 |- ( ( w e. dom F /\ z e. A ) -> ( z e. Pred ( R , A , w ) <-> z R w ) )
22	21	ancoms 469	. . . . . . . . 9 |- ( ( z e. A /\ w e. dom F ) -> ( z e. Pred ( R , A , w ) <-> z R w ) )
23	20, 22	sylan 488	. . . . . . . 8 |- ( ( z e. ( A \ dom F ) /\ w e. dom F ) -> ( z e. Pred ( R , A , w ) <-> z R w ) )
24	19, 23	mtbid 314	. . . . . . 7 |- ( ( z e. ( A \ dom F ) /\ w e. dom F ) -> -. z R w )
25	1	wfrdmss 7421	. . . . . . . . 9 |- dom F C_ A
26	25	sseli 3599	. . . . . . . 8 |- ( w e. dom F -> w e. A )
27		wfrlem10.1	. . . . . . . . . 10 |- R We A
28		weso 5105	. . . . . . . . . 10 |- ( R We A -> R Or A )
29	27, 28	ax-mp 5	. . . . . . . . 9 |- R Or A
30		solin 5058	. . . . . . . . 9 |- ( ( R Or A /\ ( w e. A /\ z e. A ) ) -> ( w R z \/ w = z \/ z R w ) )
31	29, 30	mpan 706	. . . . . . . 8 |- ( ( w e. A /\ z e. A ) -> ( w R z \/ w = z \/ z R w ) )
32	26, 20, 31	syl2anr 495	. . . . . . 7 |- ( ( z e. ( A \ dom F ) /\ w e. dom F ) -> ( w R z \/ w = z \/ z R w ) )
33	12, 24, 32	ecase23d 1436	. . . . . 6 |- ( ( z e. ( A \ dom F ) /\ w e. dom F ) -> w R z )
34		vex 3203	. . . . . . 7 |- z e. _V
35		vex 3203	. . . . . . . 8 |- w e. _V
36	35	elpred 5693	. . . . . . 7 |- ( z e. _V -> ( w e. Pred ( R , dom F , z ) <-> ( w e. dom F /\ w R z ) ) )
37	34, 36	ax-mp 5	. . . . . 6 |- ( w e. Pred ( R , dom F , z ) <-> ( w e. dom F /\ w R z ) )
38	6, 33, 37	sylanbrc 698	. . . . 5 |- ( ( z e. ( A \ dom F ) /\ w e. dom F ) -> w e. Pred ( R , dom F , z ) )
39	38	ex 450	. . . 4 |- ( z e. ( A \ dom F ) -> ( w e. dom F -> w e. Pred ( R , dom F , z ) ) )
40	39	ssrdv 3609	. . 3 |- ( z e. ( A \ dom F ) -> dom F C_ Pred ( R , dom F , z ) )
41	5, 40	eqssd 3620	. 2 |- ( z e. ( A \ dom F ) -> Pred ( R , dom F , z ) = dom F )
42	3, 41	sylan9eqr 2678	1 |- ( ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) -> Pred ( R , A , z ) = dom F )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Or wor 5034   We wwe 5072   dom cdm 5114   Predcpred 5679  wrecscwrecs 7406

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-wrecs 7407

This theorem is referenced by:  wfrlem15  7429