Theorem frrlem5e 31788

Description: Lemma for founded recursion. The domain of the union of a subset of B is closed under predecessors. (Contributed by Paul Chapman, 1-May-2012.)

Hypotheses

Ref	Expression
frrlem5.1	|- R Fr A
frrlem5.2	|- R Se A
frrlem5.3	|- B = { 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 ) = ( y G ( f |` Pred ( R , A , y ) ) ) ) ) }

Assertion

Ref	Expression
frrlem5e	|- ( C C_ B -> ( X e. dom U. C -> Pred ( R , A , X ) C_ dom U. C ) )

Distinct variable groups:   A, f, x, y   f, G, x, y   R, f, x, y   x, B

Allowed substitution hints:   B( y, f)   C( x, y, f)   X( x, y, f)

Proof of Theorem frrlem5e

Dummy variables z t w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmuni 5334	. . . 4 |- dom U. C = U_ z e. C dom z
2	1	eleq2i 2693	. . 3 |- ( X e. dom U. C <-> X e. U_ z e. C dom z )
3		eliun 4524	. . 3 |- ( X e. U_ z e. C dom z <-> E. z e. C X e. dom z )
4	2, 3	bitri 264	. 2 |- ( X e. dom U. C <-> E. z e. C X e. dom z )
5		ssel2 3598	. . . . 5 |- ( ( C C_ B /\ z e. C ) -> z e. B )
6		frrlem5.3	. . . . . . . 8 |- B = { 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 ) = ( y G ( f |` Pred ( R , A , y ) ) ) ) ) }
7	6	frrlem1 31780	. . . . . . 7 |- B = { z | E. w ( z Fn w /\ ( w C_ A /\ A. t e. w Pred ( R , A , t ) C_ w /\ A. t e. w ( z ` t ) = ( t G ( z |` Pred ( R , A , t ) ) ) ) ) }
8	7	abeq2i 2735	. . . . . 6 |- ( z e. B <-> E. w ( z Fn w /\ ( w C_ A /\ A. t e. w Pred ( R , A , t ) C_ w /\ A. t e. w ( z ` t ) = ( t G ( z |` Pred ( R , A , t ) ) ) ) ) )
9		fndm 5990	. . . . . . . . 9 |- ( z Fn w -> dom z = w )
10		predeq3 5684	. . . . . . . . . . . . 13 |- ( t = X -> Pred ( R , A , t ) = Pred ( R , A , X ) )
11	10	sseq1d 3632	. . . . . . . . . . . 12 |- ( t = X -> ( Pred ( R , A , t ) C_ w <-> Pred ( R , A , X ) C_ w ) )
12	11	rspccv 3306	. . . . . . . . . . 11 |- ( A. t e. w Pred ( R , A , t ) C_ w -> ( X e. w -> Pred ( R , A , X ) C_ w ) )
13	12	3ad2ant2 1083	. . . . . . . . . 10 |- ( ( w C_ A /\ A. t e. w Pred ( R , A , t ) C_ w /\ A. t e. w ( z ` t ) = ( t G ( z |` Pred ( R , A , t ) ) ) ) -> ( X e. w -> Pred ( R , A , X ) C_ w ) )
14		eleq2 2690	. . . . . . . . . . 11 |- ( dom z = w -> ( X e. dom z <-> X e. w ) )
15		sseq2 3627	. . . . . . . . . . 11 |- ( dom z = w -> ( Pred ( R , A , X ) C_ dom z <-> Pred ( R , A , X ) C_ w ) )
16	14, 15	imbi12d 334	. . . . . . . . . 10 |- ( dom z = w -> ( ( X e. dom z -> Pred ( R , A , X ) C_ dom z ) <-> ( X e. w -> Pred ( R , A , X ) C_ w ) ) )
17	13, 16	syl5ibr 236	. . . . . . . . 9 |- ( dom z = w -> ( ( w C_ A /\ A. t e. w Pred ( R , A , t ) C_ w /\ A. t e. w ( z ` t ) = ( t G ( z |` Pred ( R , A , t ) ) ) ) -> ( X e. dom z -> Pred ( R , A , X ) C_ dom z ) ) )
18	9, 17	syl 17	. . . . . . . 8 |- ( z Fn w -> ( ( w C_ A /\ A. t e. w Pred ( R , A , t ) C_ w /\ A. t e. w ( z ` t ) = ( t G ( z |` Pred ( R , A , t ) ) ) ) -> ( X e. dom z -> Pred ( R , A , X ) C_ dom z ) ) )
19	18	imp 445	. . . . . . 7 |- ( ( z Fn w /\ ( w C_ A /\ A. t e. w Pred ( R , A , t ) C_ w /\ A. t e. w ( z ` t ) = ( t G ( z |` Pred ( R , A , t ) ) ) ) ) -> ( X e. dom z -> Pred ( R , A , X ) C_ dom z ) )
20	19	exlimiv 1858	. . . . . 6 |- ( E. w ( z Fn w /\ ( w C_ A /\ A. t e. w Pred ( R , A , t ) C_ w /\ A. t e. w ( z ` t ) = ( t G ( z |` Pred ( R , A , t ) ) ) ) ) -> ( X e. dom z -> Pred ( R , A , X ) C_ dom z ) )
21	8, 20	sylbi 207	. . . . 5 |- ( z e. B -> ( X e. dom z -> Pred ( R , A , X ) C_ dom z ) )
22	5, 21	syl 17	. . . 4 |- ( ( C C_ B /\ z e. C ) -> ( X e. dom z -> Pred ( R , A , X ) C_ dom z ) )
23		dmeq 5324	. . . . . . . . . 10 |- ( w = z -> dom w = dom z )
24	23	sseq2d 3633	. . . . . . . . 9 |- ( w = z -> ( Pred ( R , A , X ) C_ dom w <-> Pred ( R , A , X ) C_ dom z ) )
25	24	rspcev 3309	. . . . . . . 8 |- ( ( z e. C /\ Pred ( R , A , X ) C_ dom z ) -> E. w e. C Pred ( R , A , X ) C_ dom w )
26		ssiun 4562	. . . . . . . 8 |- ( E. w e. C Pred ( R , A , X ) C_ dom w -> Pred ( R , A , X ) C_ U_ w e. C dom w )
27	25, 26	syl 17	. . . . . . 7 |- ( ( z e. C /\ Pred ( R , A , X ) C_ dom z ) -> Pred ( R , A , X ) C_ U_ w e. C dom w )
28		dmuni 5334	. . . . . . 7 |- dom U. C = U_ w e. C dom w
29	27, 28	syl6sseqr 3652	. . . . . 6 |- ( ( z e. C /\ Pred ( R , A , X ) C_ dom z ) -> Pred ( R , A , X ) C_ dom U. C )
30	29	ex 450	. . . . 5 |- ( z e. C -> ( Pred ( R , A , X ) C_ dom z -> Pred ( R , A , X ) C_ dom U. C ) )
31	30	adantl 482	. . . 4 |- ( ( C C_ B /\ z e. C ) -> ( Pred ( R , A , X ) C_ dom z -> Pred ( R , A , X ) C_ dom U. C ) )
32	22, 31	syld 47	. . 3 |- ( ( C C_ B /\ z e. C ) -> ( X e. dom z -> Pred ( R , A , X ) C_ dom U. C ) )
33	32	rexlimdva 3031	. 2 |- ( C C_ B -> ( E. z e. C X e. dom z -> Pred ( R , A , X ) C_ dom U. C ) )
34	4, 33	syl5bi 232	1 |- ( C C_ B -> ( X e. dom U. C -> Pred ( R , A , X ) C_ dom U. C ) )

Syntax hints:   -> wi 4   /\ 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   Fr wfr 5070   Se wse 5071   dom cdm 5114   |` cres 5116   Predcpred 5679   Fn wfn 5883   ` cfv 5888  (class class class)co 6650

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-ov 6653

This theorem is referenced by: (None)