Theorem bnj558 30972

Description: Technical lemma for bnj852 30991. 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
bnj558.3	|- D = ( om \ { (/) } )
bnj558.16	|- G = ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } )
bnj558.17	|- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
bnj558.18	|- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
bnj558.19	|- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
bnj558.20	|- ( ze <-> ( i e. om /\ suc i e. n /\ m = suc i ) )
bnj558.21	|- B = U_ y e. ( f ` i ) pred ( y , A , R )
bnj558.22	|- C = U_ y e. ( f ` p ) pred ( y , A , R )
bnj558.23	|- K = U_ y e. ( G ` i ) pred ( y , A , R )
bnj558.24	|- L = U_ y e. ( G ` p ) pred ( y , A , R )
bnj558.25	|- G = ( f u. { <. m , C >. } )
bnj558.28	|- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )
bnj558.29	|- ( ps' <-> A. i e. om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj558.36	|- ( ( R FrSe A /\ ta /\ si ) -> G Fn n )

Assertion

Ref	Expression
bnj558	|- ( ( R FrSe A /\ ta /\ et /\ ze ) -> ( G ` suc i ) = K )

Distinct variable groups:   A, i, p, y   y, G   R, i, p, y   f, i, p, y   i, m, p   p, ph'

Allowed substitution hints:   ta( x, y, f, i, m, n, p)   et( x, y, f, i, m, n, p)   ze( x, y, f, i, m, n, p)   si( x, y, f, i, m, n, p)   A( x, f, m, n)   B( x, y, f, i, m, n, p)   C( x, y, f, i, m, n, p)   D( x, y, f, i, m, n, p)   R( x, f, m, n)   G( x, f, i, m, n, p)   K( x, y, f, i, m, n, p)   L( x, y, f, i, m, n, p)   ph'( x, y, f, i, m, n)   ps'( x, y, f, i, m, n, p)

Proof of Theorem bnj558

Step	Hyp	Ref	Expression
1		bnj558.3	. . 3 |- D = ( om \ { (/) } )
2		bnj558.16	. . 3 |- G = ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } )
3		bnj558.17	. . 3 |- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
4		bnj558.18	. . 3 |- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
5		bnj558.19	. . 3 |- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
6		bnj558.20	. . 3 |- ( ze <-> ( i e. om /\ suc i e. n /\ m = suc i ) )
7		bnj558.21	. . 3 |- B = U_ y e. ( f ` i ) pred ( y , A , R )
8		bnj558.22	. . 3 |- C = U_ y e. ( f ` p ) pred ( y , A , R )
9		bnj558.23	. . 3 |- K = U_ y e. ( G ` i ) pred ( y , A , R )
10		bnj558.24	. . 3 |- L = U_ y e. ( G ` p ) pred ( y , A , R )
11		bnj558.25	. . 3 |- G = ( f u. { <. m , C >. } )
12		bnj558.28	. . 3 |- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )
13		bnj558.29	. . 3 |- ( ps' <-> A. i e. om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
14		bnj558.36	. . 3 |- ( ( R FrSe A /\ ta /\ si ) -> G Fn n )
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	bnj557 30971	. 2 |- ( ( R FrSe A /\ ta /\ et /\ ze ) -> ( G ` m ) = L )
16		bnj422 30781	. . . . 5 |- ( ( R FrSe A /\ ta /\ et /\ ze ) <-> ( et /\ ze /\ R FrSe A /\ ta ) )
17		bnj253 30770	. . . . 5 |- ( ( et /\ ze /\ R FrSe A /\ ta ) <-> ( ( et /\ ze ) /\ R FrSe A /\ ta ) )
18	16, 17	bitri 264	. . . 4 |- ( ( R FrSe A /\ ta /\ et /\ ze ) <-> ( ( et /\ ze ) /\ R FrSe A /\ ta ) )
19	18	simp1bi 1076	. . 3 |- ( ( R FrSe A /\ ta /\ et /\ ze ) -> ( et /\ ze ) )
20	5, 6, 9, 10, 9, 10	bnj554 30969	. . 3 |- ( ( et /\ ze ) -> ( ( G ` m ) = L <-> ( G ` suc i ) = K ) )
21	19, 20	syl 17	. 2 |- ( ( R FrSe A /\ ta /\ et /\ ze ) -> ( ( G ` m ) = L <-> ( G ` suc i ) = K ) )
22	15, 21	mpbid 222	1 |- ( ( R FrSe A /\ ta /\ et /\ ze ) -> ( G ` suc i ) = K )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   u. cun 3572   (/)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   FrSe w-bnj15 30758

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  ax-reg 8497

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-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-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-id 5024  df-eprel 5029  df-fr 5073  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-bnj17 30753

This theorem is referenced by:  bnj571  30976