Theorem wfrlem8 7422

Description: Lemma for well-founded recursion. Compute the prececessor class for an R minimal element of ( A \ dom F ). (Contributed by Scott Fenton, 21-Apr-2011.)

Hypothesis

Ref	Expression
wfrlem6.1	|- F = wrecs ( R , A , G )

Assertion

Ref	Expression
wfrlem8	|- ( Pred ( R , ( A \ dom F ) , X ) = (/) <-> Pred ( R , A , X ) = Pred ( R , dom F , X ) )

Proof of Theorem wfrlem8

Step	Hyp	Ref	Expression
1		wfrlem6.1	. . . . 5 |- F = wrecs ( R , A , G )
2	1	wfrdmss 7421	. . . 4 |- dom F C_ A
3		predpredss 5686	. . . 4 |- ( dom F C_ A -> Pred ( R , dom F , X ) C_ Pred ( R , A , X ) )
4	2, 3	ax-mp 5	. . 3 |- Pred ( R , dom F , X ) C_ Pred ( R , A , X )
5	4	biantru 526	. 2 |- ( Pred ( R , A , X ) C_ Pred ( R , dom F , X ) <-> ( Pred ( R , A , X ) C_ Pred ( R , dom F , X ) /\ Pred ( R , dom F , X ) C_ Pred ( R , A , X ) ) )
6		preddif 5705	. . . 4 |- Pred ( R , ( A \ dom F ) , X ) = ( Pred ( R , A , X ) \ Pred ( R , dom F , X ) )
7	6	eqeq1i 2627	. . 3 |- ( Pred ( R , ( A \ dom F ) , X ) = (/) <-> ( Pred ( R , A , X ) \ Pred ( R , dom F , X ) ) = (/) )
8		ssdif0 3942	. . 3 |- ( Pred ( R , A , X ) C_ Pred ( R , dom F , X ) <-> ( Pred ( R , A , X ) \ Pred ( R , dom F , X ) ) = (/) )
9	7, 8	bitr4i 267	. 2 |- ( Pred ( R , ( A \ dom F ) , X ) = (/) <-> Pred ( R , A , X ) C_ Pred ( R , dom F , X ) )
10		eqss 3618	. 2 |- ( Pred ( R , A , X ) = Pred ( R , dom F , X ) <-> ( Pred ( R , A , X ) C_ Pred ( R , dom F , X ) /\ Pred ( R , dom F , X ) C_ Pred ( R , A , X ) ) )
11	5, 9, 10	3bitr4i 292	1 |- ( Pred ( R , ( A \ dom F ) , X ) = (/) <-> Pred ( R , A , X ) = Pred ( R , dom F , X ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   \ cdif 3571   C_ wss 3574   (/)c0 3915   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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  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-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-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:  wfrlem10  7424