Theorem bnj1014 29332

Description: Technical lemma for bnj69 29380. 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
bnj1014.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj1014.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj1014.13	|- D = ( om \ { (/) } )
bnj1014.14	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }

Assertion

Ref	Expression
bnj1014	|- ( ( g e. B /\ j e. dom g ) -> ( g ` j ) C_ trCl ( X , A , R ) )

Distinct variable groups:   A, f, i, n, y   D, i   R, f, i, n, y   f, X, i, n, y   f, g, i   i, j   ph, i

Allowed substitution hints:   ph( y, f, g, j, n)   ps( y, f, g, i, j, n)   A( g, j)   B( y, f, g, i, j, n)   D( y, f, g, j, n)   R( g, j)   X( g, j)

Proof of Theorem bnj1014

Step	Hyp	Ref	Expression
1		bnj1014.14	. . . . . . 7 |- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
2		nfcv 2564	. . . . . . . . 9 |- F/_ i D
3		bnj1014.1	. . . . . . . . . . 11 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
4		bnj1014.2	. . . . . . . . . . 11 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
5	3, 4	bnj911 29304	. . . . . . . . . 10 |- ( ( f Fn n /\ ph /\ ps ) -> A. i ( f Fn n /\ ph /\ ps ) )
6	5	nfi 1644	. . . . . . . . 9 |- F/ i ( f Fn n /\ ph /\ ps )
7	2, 6	nfrex 2866	. . . . . . . 8 |- F/ i E. n e. D ( f Fn n /\ ph /\ ps )
8	7	nfab 2568	. . . . . . 7 |- F/_ i { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
9	1, 8	nfcxfr 2562	. . . . . 6 |- F/_ i B
10	9	nfcri 2557	. . . . 5 |- F/ i g e. B
11		nfv 1728	. . . . 5 |- F/ i j e. dom g
12	10, 11	nfan 1956	. . . 4 |- F/ i ( g e. B /\ j e. dom g )
13		nfv 1728	. . . 4 |- F/ i ( g ` j ) C_ trCl ( X , A , R )
14	12, 13	nfim 1948	. . 3 |- F/ i ( ( g e. B /\ j e. dom g ) -> ( g ` j ) C_ trCl ( X , A , R ) )
15	14	nfri 1898	. 2 |- ( ( ( g e. B /\ j e. dom g ) -> ( g ` j ) C_ trCl ( X , A , R ) ) -> A. i ( ( g e. B /\ j e. dom g ) -> ( g ` j ) C_ trCl ( X , A , R ) ) )
16		eleq1 2474	. . . . . 6 |- ( j = i -> ( j e. dom g <-> i e. dom g ) )
17	16	anbi2d 702	. . . . 5 |- ( j = i -> ( ( g e. B /\ j e. dom g ) <-> ( g e. B /\ i e. dom g ) ) )
18		fveq2 5848	. . . . . 6 |- ( j = i -> ( g ` j ) = ( g ` i ) )
19	18	sseq1d 3468	. . . . 5 |- ( j = i -> ( ( g ` j ) C_ trCl ( X , A , R ) <-> ( g ` i ) C_ trCl ( X , A , R ) ) )
20	17, 19	imbi12d 318	. . . 4 |- ( j = i -> ( ( ( g e. B /\ j e. dom g ) -> ( g ` j ) C_ trCl ( X , A , R ) ) <-> ( ( g e. B /\ i e. dom g ) -> ( g ` i ) C_ trCl ( X , A , R ) ) ) )
21	20	equcoms 1819	. . 3 |- ( i = j -> ( ( ( g e. B /\ j e. dom g ) -> ( g ` j ) C_ trCl ( X , A , R ) ) <-> ( ( g e. B /\ i e. dom g ) -> ( g ` i ) C_ trCl ( X , A , R ) ) ) )
22	1	bnj1317 29194	. . . . . . 7 |- ( g e. B -> A. f g e. B )
23	22	nfi 1644	. . . . . 6 |- F/ f g e. B
24		nfv 1728	. . . . . 6 |- F/ f i e. dom g
25	23, 24	nfan 1956	. . . . 5 |- F/ f ( g e. B /\ i e. dom g )
26		nfv 1728	. . . . 5 |- F/ f ( g ` i ) C_ trCl ( X , A , R )
27	25, 26	nfim 1948	. . . 4 |- F/ f ( ( g e. B /\ i e. dom g ) -> ( g ` i ) C_ trCl ( X , A , R ) )
28		eleq1 2474	. . . . . 6 |- ( f = g -> ( f e. B <-> g e. B ) )
29		dmeq 5023	. . . . . . 7 |- ( f = g -> dom f = dom g )
30	29	eleq2d 2472	. . . . . 6 |- ( f = g -> ( i e. dom f <-> i e. dom g ) )
31	28, 30	anbi12d 709	. . . . 5 |- ( f = g -> ( ( f e. B /\ i e. dom f ) <-> ( g e. B /\ i e. dom g ) ) )
32		fveq1 5847	. . . . . 6 |- ( f = g -> ( f ` i ) = ( g ` i ) )
33	32	sseq1d 3468	. . . . 5 |- ( f = g -> ( ( f ` i ) C_ trCl ( X , A , R ) <-> ( g ` i ) C_ trCl ( X , A , R ) ) )
34	31, 33	imbi12d 318	. . . 4 |- ( f = g -> ( ( ( f e. B /\ i e. dom f ) -> ( f ` i ) C_ trCl ( X , A , R ) ) <-> ( ( g e. B /\ i e. dom g ) -> ( g ` i ) C_ trCl ( X , A , R ) ) ) )
35		ssiun2 4313	. . . . 5 |- ( i e. dom f -> ( f ` i ) C_ U_ i e. dom f ( f ` i ) )
36		ssiun2 4313	. . . . . 6 |- ( f e. B -> U_ i e. dom f ( f ` i ) C_ U_ f e. B U_ i e. dom f ( f ` i ) )
37		bnj1014.13	. . . . . . 7 |- D = ( om \ { (/) } )
38	3, 4, 37, 1	bnj882 29298	. . . . . 6 |- trCl ( X , A , R ) = U_ f e. B U_ i e. dom f ( f ` i )
39	36, 38	syl6sseqr 3488	. . . . 5 |- ( f e. B -> U_ i e. dom f ( f ` i ) C_ trCl ( X , A , R ) )
40	35, 39	sylan9ssr 3455	. . . 4 |- ( ( f e. B /\ i e. dom f ) -> ( f ` i ) C_ trCl ( X , A , R ) )
41	27, 34, 40	chvar 2040	. . 3 |- ( ( g e. B /\ i e. dom g ) -> ( g ` i ) C_ trCl ( X , A , R ) )
42	21, 41	spei 2039	. 2 |- E. i ( ( g e. B /\ j e. dom g ) -> ( g ` j ) C_ trCl ( X , A , R ) )
43	15, 42	bnj1131 29160	1 |- ( ( g e. B /\ j e. dom g ) -> ( g ` j ) C_ trCl ( X , A , R ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   {cab 2387   A.wral 2753   E.wrex 2754   \ cdif 3410   C_ wss 3413   (/)c0 3737   {csn 3971   U_ciun 4270   dom cdm 4822   suc csuc 5411   Fn wfn 5563   ` cfv 5568   omcom 6682   predc-bnj14 29054   trClc-bnj18 29060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-dm 4832  df-iota 5532  df-fv 5576  df-bnj18 29061

This theorem is referenced by:  bnj1015  29333