Theorem wfrdmcl 7423

Description: Given F = wrecs ( R , A , X ) /\ X e. dom F, then its predecessor class is a subset of dom F. (Contributed by Scott Fenton, 21-Apr-2011.)

Hypothesis

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

Assertion

Ref	Expression
wfrdmcl	|- ( X e. dom F -> Pred ( R , A , X ) C_ dom F )

Proof of Theorem wfrdmcl

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

Step	Hyp	Ref	Expression
1		wfrlem6.1	. . . . . . . 8 |- F = wrecs ( R , A , G )
2		df-wrecs 7407	. . . . . . . 8 |- 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 ) ) ) ) }
3	1, 2	eqtri 2644	. . . . . . 7 |- 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 ) ) ) ) }
4	3	dmeqi 5325	. . . . . 6 |- dom F = dom 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 ) ) ) ) }
5		dmuni 5334	. . . . . 6 |- dom 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 ) ) ) ) } = U_ g 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 ) ) ) ) } dom g
6	4, 5	eqtri 2644	. . . . 5 |- dom F = U_ g 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 ) ) ) ) } dom g
7	6	eleq2i 2693	. . . 4 |- ( X e. dom F <-> X e. U_ g 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 ) ) ) ) } dom g )
8		eliun 4524	. . . 4 |- ( X e. U_ g 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 ) ) ) ) } dom g <-> E. g 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 ) ) ) ) } X e. dom g )
9	7, 8	bitri 264	. . 3 |- ( X e. dom F <-> E. g 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 ) ) ) ) } X e. dom g )
10		eqid 2622	. . . . . . . 8 |- { 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 ) ) ) ) } = { 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 ) ) ) ) }
11	10	wfrlem1 7414	. . . . . . 7 |- { 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 ) ) ) ) } = { g | E. z ( g Fn z /\ ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z ) /\ A. w e. z ( g ` w ) = ( G ` ( g |` Pred ( R , A , w ) ) ) ) }
12	11	abeq2i 2735	. . . . . 6 |- ( g 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 ) ) ) ) } <-> E. z ( g Fn z /\ ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z ) /\ A. w e. z ( g ` w ) = ( G ` ( g |` Pred ( R , A , w ) ) ) ) )
13		predeq3 5684	. . . . . . . . . . . . 13 |- ( w = X -> Pred ( R , A , w ) = Pred ( R , A , X ) )
14	13	sseq1d 3632	. . . . . . . . . . . 12 |- ( w = X -> ( Pred ( R , A , w ) C_ z <-> Pred ( R , A , X ) C_ z ) )
15	14	rspccv 3306	. . . . . . . . . . 11 |- ( A. w e. z Pred ( R , A , w ) C_ z -> ( X e. z -> Pred ( R , A , X ) C_ z ) )
16	15	adantl 482	. . . . . . . . . 10 |- ( ( g Fn z /\ A. w e. z Pred ( R , A , w ) C_ z ) -> ( X e. z -> Pred ( R , A , X ) C_ z ) )
17		fndm 5990	. . . . . . . . . . . . 13 |- ( g Fn z -> dom g = z )
18	17	eleq2d 2687	. . . . . . . . . . . 12 |- ( g Fn z -> ( X e. dom g <-> X e. z ) )
19	17	sseq2d 3633	. . . . . . . . . . . 12 |- ( g Fn z -> ( Pred ( R , A , X ) C_ dom g <-> Pred ( R , A , X ) C_ z ) )
20	18, 19	imbi12d 334	. . . . . . . . . . 11 |- ( g Fn z -> ( ( X e. dom g -> Pred ( R , A , X ) C_ dom g ) <-> ( X e. z -> Pred ( R , A , X ) C_ z ) ) )
21	20	adantr 481	. . . . . . . . . 10 |- ( ( g Fn z /\ A. w e. z Pred ( R , A , w ) C_ z ) -> ( ( X e. dom g -> Pred ( R , A , X ) C_ dom g ) <-> ( X e. z -> Pred ( R , A , X ) C_ z ) ) )
22	16, 21	mpbird 247	. . . . . . . . 9 |- ( ( g Fn z /\ A. w e. z Pred ( R , A , w ) C_ z ) -> ( X e. dom g -> Pred ( R , A , X ) C_ dom g ) )
23	22	adantrl 752	. . . . . . . 8 |- ( ( g Fn z /\ ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z ) ) -> ( X e. dom g -> Pred ( R , A , X ) C_ dom g ) )
24	23	3adant3 1081	. . . . . . 7 |- ( ( g Fn z /\ ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z ) /\ A. w e. z ( g ` w ) = ( G ` ( g |` Pred ( R , A , w ) ) ) ) -> ( X e. dom g -> Pred ( R , A , X ) C_ dom g ) )
25	24	exlimiv 1858	. . . . . 6 |- ( E. z ( g Fn z /\ ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z ) /\ A. w e. z ( g ` w ) = ( G ` ( g |` Pred ( R , A , w ) ) ) ) -> ( X e. dom g -> Pred ( R , A , X ) C_ dom g ) )
26	12, 25	sylbi 207	. . . . 5 |- ( g 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 ) ) ) ) } -> ( X e. dom g -> Pred ( R , A , X ) C_ dom g ) )
27	26	reximia 3009	. . . 4 |- ( E. g 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 ) ) ) ) } X e. dom g -> E. g 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 ) ) ) ) } Pred ( R , A , X ) C_ dom g )
28		ssiun 4562	. . . 4 |- ( E. g 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 ) ) ) ) } Pred ( R , A , X ) C_ dom g -> Pred ( R , A , X ) C_ U_ g 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 ) ) ) ) } dom g )
29	27, 28	syl 17	. . 3 |- ( E. g 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 ) ) ) ) } X e. dom g -> Pred ( R , A , X ) C_ U_ g 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 ) ) ) ) } dom g )
30	9, 29	sylbi 207	. 2 |- ( X e. dom F -> Pred ( R , A , X ) C_ U_ g 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 ) ) ) ) } dom g )
31	30, 6	syl6sseqr 3652	1 |- ( X e. dom F -> Pred ( R , A , X ) C_ dom F )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   C_ wss 3574   U.cuni 4436   U_ciun 4520   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-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  wfrlem14  7428  wfrlem15  7429