Theorem bnj999 31027

Description: Technical lemma for bnj69 31078. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj999.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj999.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj999.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj999.7	|- ( ph' <-> [. p / n ]. ph )
bnj999.8	|- ( ps' <-> [. p / n ]. ps )
bnj999.9	|- ( ch' <-> [. p / n ]. ch )
bnj999.10	|- ( ph" <-> [. G / f ]. ph' )
bnj999.11	|- ( ps" <-> [. G / f ]. ps' )
bnj999.12	|- ( ch" <-> [. G / f ]. ch' )
bnj999.15	|- C = U_ y e. ( f ` m ) pred ( y , A , R )
bnj999.16	|- G = ( f u. { <. n , C >. } )

Assertion

Ref	Expression
bnj999	|- ( ( ch" /\ i e. om /\ suc i e. p /\ y e. ( G ` i ) ) -> pred ( y , A , R ) C_ ( G ` suc i ) )

Distinct variable groups:   f, i, n, y   A, f, n   D, f, n   i, G   R, f, n   f, X, n   f, p, i, n

Allowed substitution hints:   ph( y, f, i, m, n, p)   ps( y, f, i, m, n, p)   ch( y, f, i, m, n, p)   A( y, i, m, p)   C( y, f, i, m, n, p)   D( y, i, m, p)   R( y, i, m, p)   G( y, f, m, n, p)   X( y, i, m, p)   ph'( y, f, i, m, n, p)   ps'( y, f, i, m, n, p)   ch'( y, f, i, m, n, p)   ph"( y, f, i, m, n, p)   ps"( y, f, i, m, n, p)   ch"( y, f, i, m, n, p)

Proof of Theorem bnj999

Step	Hyp	Ref	Expression
1		bnj999.3	. . . . . . 7 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
2		bnj999.7	. . . . . . 7 |- ( ph' <-> [. p / n ]. ph )
3		bnj999.8	. . . . . . 7 |- ( ps' <-> [. p / n ]. ps )
4		bnj999.9	. . . . . . 7 |- ( ch' <-> [. p / n ]. ch )
5		vex 3203	. . . . . . 7 |- p e. _V
6	1, 2, 3, 4, 5	bnj919 30837	. . . . . 6 |- ( ch' <-> ( p e. D /\ f Fn p /\ ph' /\ ps' ) )
7		bnj999.10	. . . . . 6 |- ( ph" <-> [. G / f ]. ph' )
8		bnj999.11	. . . . . 6 |- ( ps" <-> [. G / f ]. ps' )
9		bnj999.12	. . . . . 6 |- ( ch" <-> [. G / f ]. ch' )
10		bnj999.16	. . . . . . 7 |- G = ( f u. { <. n , C >. } )
11	10	bnj918 30836	. . . . . 6 |- G e. _V
12	6, 7, 8, 9, 11	bnj976 30848	. . . . 5 |- ( ch" <-> ( p e. D /\ G Fn p /\ ph" /\ ps" ) )
13	12	bnj1254 30880	. . . 4 |- ( ch" -> ps" )
14	13	anim1i 592	. . 3 |- ( ( ch" /\ ( i e. om /\ suc i e. p /\ y e. ( G ` i ) ) ) -> ( ps" /\ ( i e. om /\ suc i e. p /\ y e. ( G ` i ) ) ) )
15		bnj252 30769	. . 3 |- ( ( ch" /\ i e. om /\ suc i e. p /\ y e. ( G ` i ) ) <-> ( ch" /\ ( i e. om /\ suc i e. p /\ y e. ( G ` i ) ) ) )
16		bnj252 30769	. . 3 |- ( ( ps" /\ i e. om /\ suc i e. p /\ y e. ( G ` i ) ) <-> ( ps" /\ ( i e. om /\ suc i e. p /\ y e. ( G ` i ) ) ) )
17	14, 15, 16	3imtr4i 281	. 2 |- ( ( ch" /\ i e. om /\ suc i e. p /\ y e. ( G ` i ) ) -> ( ps" /\ i e. om /\ suc i e. p /\ y e. ( G ` i ) ) )
18		ssiun2 4563	. . . 4 |- ( y e. ( G ` i ) -> pred ( y , A , R ) C_ U_ y e. ( G ` i ) pred ( y , A , R ) )
19	18	bnj708 30826	. . 3 |- ( ( ps" /\ i e. om /\ suc i e. p /\ y e. ( G ` i ) ) -> pred ( y , A , R ) C_ U_ y e. ( G ` i ) pred ( y , A , R ) )
20		3simpa 1058	. . . . . 6 |- ( ( ps" /\ i e. om /\ suc i e. p ) -> ( ps" /\ i e. om ) )
21	20	ancomd 467	. . . . 5 |- ( ( ps" /\ i e. om /\ suc i e. p ) -> ( i e. om /\ ps" ) )
22		simp3 1063	. . . . 5 |- ( ( ps" /\ i e. om /\ suc i e. p ) -> suc i e. p )
23		bnj999.2	. . . . . . . 8 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
24	23, 3, 5	bnj539 30961	. . . . . . 7 |- ( ps' <-> A. i e. om ( suc i e. p -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
25		bnj999.15	. . . . . . 7 |- C = U_ y e. ( f ` m ) pred ( y , A , R )
26	24, 8, 25, 10	bnj965 31012	. . . . . 6 |- ( ps" <-> A. i e. om ( suc i e. p -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
27	26	bnj228 30803	. . . . 5 |- ( ( i e. om /\ ps" ) -> ( suc i e. p -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
28	21, 22, 27	sylc 65	. . . 4 |- ( ( ps" /\ i e. om /\ suc i e. p ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
29	28	bnj721 30827	. . 3 |- ( ( ps" /\ i e. om /\ suc i e. p /\ y e. ( G ` i ) ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
30	19, 29	sseqtr4d 3642	. 2 |- ( ( ps" /\ i e. om /\ suc i e. p /\ y e. ( G ` i ) ) -> pred ( y , A , R ) C_ ( G ` suc i ) )
31	17, 30	syl 17	1 |- ( ( ch" /\ i e. om /\ suc i e. p /\ y e. ( G ` i ) ) -> pred ( y , A , R ) C_ ( G ` suc i ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   [.wsbc 3435   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U_ciun 4520   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   /\ w-bnj17 30752   predc-bnj14 30754

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-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

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-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-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-bnj17 30753

This theorem is referenced by:  bnj1006  31029