Theorem bnj967 29751

Description: Technical lemma for bnj69 29814. 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
bnj967.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj967.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj967.10	|- D = ( om \ { (/) } )
bnj967.12	|- C = U_ y e. ( f ` m ) pred ( y , A , R )
bnj967.13	|- G = ( f u. { <. n , C >. } )
bnj967.44	|- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> C e. _V )

Assertion

Ref	Expression
bnj967	|- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )

Distinct variable groups:   y, f   y, i   y, 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, f, i, m, n, p)   C( y, f, i, m, n, p)   D( y, f, i, m, n, p)   R( y, f, i, m, n, p)   G( y, f, i, m, n, p)   X( y, f, i, m, n, p)

Proof of Theorem bnj967

Step	Hyp	Ref	Expression
1		bnj967.44	. . . . . . 7 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> C e. _V )
2	1	3adant3 1025	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> C e. _V )
3		bnj967.3	. . . . . . . . 9 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4	3	bnj1235 29611	. . . . . . . 8 |- ( ch -> f Fn n )
5	4	3ad2ant1 1026	. . . . . . 7 |- ( ( ch /\ n = suc m /\ p = suc n ) -> f Fn n )
6	5	3ad2ant2 1027	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> f Fn n )
7		simp23 1040	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> p = suc n )
8		simp3 1007	. . . . . . 7 |- ( ( i e. om /\ suc i e. p /\ suc i e. n ) -> suc i e. n )
9	8	3ad2ant3 1028	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> suc i e. n )
10	2, 6, 7, 9	bnj951 29582	. . . . 5 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> ( C e. _V /\ f Fn n /\ p = suc n /\ suc i e. n ) )
11		bnj967.10	. . . . . . . . . 10 |- D = ( om \ { (/) } )
12	11	bnj923 29574	. . . . . . . . 9 |- ( n e. D -> n e. om )
13	3, 12	bnj769 29568	. . . . . . . 8 |- ( ch -> n e. om )
14	13	3ad2ant1 1026	. . . . . . 7 |- ( ( ch /\ n = suc m /\ p = suc n ) -> n e. om )
15	14, 8	bnj240 29499	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> ( n e. om /\ suc i e. n ) )
16		nnord 6710	. . . . . . . 8 |- ( n e. om -> Ord n )
17		ordtr 5452	. . . . . . . 8 |- ( Ord n -> Tr n )
18	16, 17	syl 17	. . . . . . 7 |- ( n e. om -> Tr n )
19		trsuc 5522	. . . . . . 7 |- ( ( Tr n /\ suc i e. n ) -> i e. n )
20	18, 19	sylan 473	. . . . . 6 |- ( ( n e. om /\ suc i e. n ) -> i e. n )
21	15, 20	syl 17	. . . . 5 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> i e. n )
22		bnj658 29556	. . . . . . 7 |- ( ( C e. _V /\ f Fn n /\ p = suc n /\ suc i e. n ) -> ( C e. _V /\ f Fn n /\ p = suc n ) )
23	22	anim1i 570	. . . . . 6 |- ( ( ( C e. _V /\ f Fn n /\ p = suc n /\ suc i e. n ) /\ i e. n ) -> ( ( C e. _V /\ f Fn n /\ p = suc n ) /\ i e. n ) )
24		df-bnj17 29487	. . . . . 6 |- ( ( C e. _V /\ f Fn n /\ p = suc n /\ i e. n ) <-> ( ( C e. _V /\ f Fn n /\ p = suc n ) /\ i e. n ) )
25	23, 24	sylibr 215	. . . . 5 |- ( ( ( C e. _V /\ f Fn n /\ p = suc n /\ suc i e. n ) /\ i e. n ) -> ( C e. _V /\ f Fn n /\ p = suc n /\ i e. n ) )
26	10, 21, 25	syl2anc 665	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> ( C e. _V /\ f Fn n /\ p = suc n /\ i e. n ) )
27		bnj967.13	. . . . 5 |- G = ( f u. { <. n , C >. } )
28	27	bnj945 29580	. . . 4 |- ( ( C e. _V /\ f Fn n /\ p = suc n /\ i e. n ) -> ( G ` i ) = ( f ` i ) )
29	26, 28	syl 17	. . 3 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> ( G ` i ) = ( f ` i ) )
30	27	bnj945 29580	. . . 4 |- ( ( C e. _V /\ f Fn n /\ p = suc n /\ suc i e. n ) -> ( G ` suc i ) = ( f ` suc i ) )
31	10, 30	syl 17	. . 3 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> ( G ` suc i ) = ( f ` suc i ) )
32		3simpb 1003	. . . 4 |- ( ( i e. om /\ suc i e. p /\ suc i e. n ) -> ( i e. om /\ suc i e. n ) )
33	32	3ad2ant3 1028	. . 3 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> ( i e. om /\ suc i e. n ) )
34	3	bnj1254 29616	. . . . 5 |- ( ch -> ps )
35	34	3ad2ant1 1026	. . . 4 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ps )
36	35	3ad2ant2 1027	. . 3 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> ps )
37	29, 31, 33, 36	bnj951 29582	. 2 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> ( ( G ` i ) = ( f ` i ) /\ ( G ` suc i ) = ( f ` suc i ) /\ ( i e. om /\ suc i e. n ) /\ ps ) )
38		bnj967.2	. . 3 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
39		bnj967.12	. . . 4 |- C = U_ y e. ( f ` m ) pred ( y , A , R )
40	39, 27	bnj958 29746	. . 3 |- ( ( G ` i ) = ( f ` i ) -> A. y ( G ` i ) = ( f ` i ) )
41	38, 40	bnj953 29745	. 2 |- ( ( ( G ` i ) = ( f ` i ) /\ ( G ` suc i ) = ( f ` suc i ) /\ ( i e. om /\ suc i e. n ) /\ ps ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
42	37, 41	syl 17	1 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1868   A.wral 2775   _Vcvv 3081   \ cdif 3433   u. cun 3434   (/)c0 3761   {csn 3996   <.cop 4002   U_ciun 4296   Tr wtr 4515   Ord word 5437   suc csuc 5440   Fn wfn 5592   ` cfv 5597   omcom 6702   /\ w-bnj17 29486   predc-bnj14 29488   FrSe w-bnj15 29492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656  ax-un 6593  ax-reg 8109

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-res 4861  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-fv 5605  df-om 6703  df-bnj17 29487

This theorem is referenced by:  bnj910  29754