Theorem wfrlem16 7430

Description: Lemma for well-founded recursion. If z is R minimal in ( A \ dom F ), then C is acceptable and thus a subset of F, but dom C is bigger than dom F. Thus, z cannot be minimal, so ( A \ dom F ) must be empty, and (due to wfrdmss 7421), dom F = A. (Contributed by Scott Fenton, 21-Apr-2011.)

Hypotheses

Ref	Expression
wfrlem13.1	|- R We A
wfrlem13.2	|- R Se A
wfrlem13.3	|- F = wrecs ( R , A , G )
wfrlem13.4	|- C = ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } )

Assertion

Ref	Expression
wfrlem16	|- dom F = A

Distinct variable groups:   z, A   z, F   z, R

Allowed substitution hints:   C( z)   G( z)

Proof of Theorem wfrlem16

Dummy variables f x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wfrlem13.3	. . 3 |- F = wrecs ( R , A , G )
2	1	wfrdmss 7421	. 2 |- dom F C_ A
3		eldifn 3733	. . . . . 6 |- ( z e. ( A \ dom F ) -> -. z e. dom F )
4		ssun2 3777	. . . . . . . . 9 |- { z } C_ ( dom F u. { z } )
5		vsnid 4209	. . . . . . . . 9 |- z e. { z }
6	4, 5	sselii 3600	. . . . . . . 8 |- z e. ( dom F u. { z } )
7		wfrlem13.4	. . . . . . . . . 10 |- C = ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } )
8	7	dmeqi 5325	. . . . . . . . 9 |- dom C = dom ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } )
9		dmun 5331	. . . . . . . . 9 |- dom ( F u. { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) = ( dom F u. dom { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } )
10		fvex 6201	. . . . . . . . . . 11 |- ( G ` ( F |` Pred ( R , A , z ) ) ) e. _V
11	10	dmsnop 5609	. . . . . . . . . 10 |- dom { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } = { z }
12	11	uneq2i 3764	. . . . . . . . 9 |- ( dom F u. dom { <. z , ( G ` ( F |` Pred ( R , A , z ) ) ) >. } ) = ( dom F u. { z } )
13	8, 9, 12	3eqtri 2648	. . . . . . . 8 |- dom C = ( dom F u. { z } )
14	6, 13	eleqtrri 2700	. . . . . . 7 |- z e. dom C
15		wfrlem13.1	. . . . . . . . . . . 12 |- R We A
16		wfrlem13.2	. . . . . . . . . . . 12 |- R Se A
17	15, 16, 1, 7	wfrlem15 7429	. . . . . . . . . . 11 |- ( ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) -> C e. { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( G ` ( f |` Pred ( R , A , y ) ) ) ) } )
18		elssuni 4467	. . . . . . . . . . 11 |- ( C e. { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( G ` ( f |` Pred ( R , A , y ) ) ) ) } -> C C_ U. { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( G ` ( f |` Pred ( R , A , y ) ) ) ) } )
19	17, 18	syl 17	. . . . . . . . . 10 |- ( ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) -> C C_ U. { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( G ` ( f |` Pred ( R , A , y ) ) ) ) } )
20		df-wrecs 7407	. . . . . . . . . . 11 |- wrecs ( R , A , G ) = U. { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( G ` ( f |` Pred ( R , A , y ) ) ) ) }
21	1, 20	eqtri 2644	. . . . . . . . . 10 |- F = U. { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( G ` ( f |` Pred ( R , A , y ) ) ) ) }
22	19, 21	syl6sseqr 3652	. . . . . . . . 9 |- ( ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) -> C C_ F )
23		dmss 5323	. . . . . . . . 9 |- ( C C_ F -> dom C C_ dom F )
24	22, 23	syl 17	. . . . . . . 8 |- ( ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) -> dom C C_ dom F )
25	24	sseld 3602	. . . . . . 7 |- ( ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) -> ( z e. dom C -> z e. dom F ) )
26	14, 25	mpi 20	. . . . . 6 |- ( ( z e. ( A \ dom F ) /\ Pred ( R , ( A \ dom F ) , z ) = (/) ) -> z e. dom F )
27	3, 26	mtand 691	. . . . 5 |- ( z e. ( A \ dom F ) -> -. Pred ( R , ( A \ dom F ) , z ) = (/) )
28	27	nrex 3000	. . . 4 |- -. E. z e. ( A \ dom F ) Pred ( R , ( A \ dom F ) , z ) = (/)
29		df-ne 2795	. . . . 5 |- ( ( A \ dom F ) =/= (/) <-> -. ( A \ dom F ) = (/) )
30		difss 3737	. . . . . 6 |- ( A \ dom F ) C_ A
31	15, 16	tz6.26i 5712	. . . . . 6 |- ( ( ( A \ dom F ) C_ A /\ ( A \ dom F ) =/= (/) ) -> E. z e. ( A \ dom F ) Pred ( R , ( A \ dom F ) , z ) = (/) )
32	30, 31	mpan 706	. . . . 5 |- ( ( A \ dom F ) =/= (/) -> E. z e. ( A \ dom F ) Pred ( R , ( A \ dom F ) , z ) = (/) )
33	29, 32	sylbir 225	. . . 4 |- ( -. ( A \ dom F ) = (/) -> E. z e. ( A \ dom F ) Pred ( R , ( A \ dom F ) , z ) = (/) )
34	28, 33	mt3 192	. . 3 |- ( A \ dom F ) = (/)
35		ssdif0 3942	. . 3 |- ( A C_ dom F <-> ( A \ dom F ) = (/) )
36	34, 35	mpbir 221	. 2 |- A C_ dom F
37	2, 36	eqssi 3619	1 |- dom F = A

Syntax hints:   -. wn 3   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U.cuni 4436   Se wse 5071   We wwe 5072   dom cdm 5114   |` cres 5116   Predcpred 5679   Fn wfn 5883   ` cfv 5888  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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  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-rmo 2920  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-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-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  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-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-wrecs 7407

This theorem is referenced by:  wfr1  7433  wfr2  7434