Theorem bnj1014 29843

Description: Technical lemma for bnj69 29891. 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 2612	. . . . . . . . 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 29815	. . . . . . . . . 10 |- ( ( f Fn n /\ ph /\ ps ) -> A. i ( f Fn n /\ ph /\ ps ) )
6	5	nfi 1682	. . . . . . . . 9 |- F/ i ( f Fn n /\ ph /\ ps )
7	2, 6	nfrex 2848	. . . . . . . 8 |- F/ i E. n e. D ( f Fn n /\ ph /\ ps )
8	7	nfab 2616	. . . . . . 7 |- F/_ i { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
9	1, 8	nfcxfr 2610	. . . . . 6 |- F/_ i B
10	9	nfcri 2606	. . . . 5 |- F/ i g e. B
11		nfv 1769	. . . . 5 |- F/ i j e. dom g
12	10, 11	nfan 2031	. . . 4 |- F/ i ( g e. B /\ j e. dom g )
13		nfv 1769	. . . 4 |- F/ i ( g ` j ) C_ trCl ( X , A , R )
14	12, 13	nfim 2023	. . 3 |- F/ i ( ( g e. B /\ j e. dom g ) -> ( g ` j ) C_ trCl ( X , A , R ) )
15	14	nfri 1972	. 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 2537	. . . . . 6 |- ( j = i -> ( j e. dom g <-> i e. dom g ) )
17	16	anbi2d 718	. . . . 5 |- ( j = i -> ( ( g e. B /\ j e. dom g ) <-> ( g e. B /\ i e. dom g ) ) )
18		fveq2 5879	. . . . . 6 |- ( j = i -> ( g ` j ) = ( g ` i ) )
19	18	sseq1d 3445	. . . . 5 |- ( j = i -> ( ( g ` j ) C_ trCl ( X , A , R ) <-> ( g ` i ) C_ trCl ( X , A , R ) ) )
20	17, 19	imbi12d 327	. . . 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 1872	. . 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 29705	. . . . . . 7 |- ( g e. B -> A. f g e. B )
23	22	nfi 1682	. . . . . 6 |- F/ f g e. B
24		nfv 1769	. . . . . 6 |- F/ f i e. dom g
25	23, 24	nfan 2031	. . . . 5 |- F/ f ( g e. B /\ i e. dom g )
26		nfv 1769	. . . . 5 |- F/ f ( g ` i ) C_ trCl ( X , A , R )
27	25, 26	nfim 2023	. . . 4 |- F/ f ( ( g e. B /\ i e. dom g ) -> ( g ` i ) C_ trCl ( X , A , R ) )
28		eleq1 2537	. . . . . 6 |- ( f = g -> ( f e. B <-> g e. B ) )
29		dmeq 5040	. . . . . . 7 |- ( f = g -> dom f = dom g )
30	29	eleq2d 2534	. . . . . 6 |- ( f = g -> ( i e. dom f <-> i e. dom g ) )
31	28, 30	anbi12d 725	. . . . 5 |- ( f = g -> ( ( f e. B /\ i e. dom f ) <-> ( g e. B /\ i e. dom g ) ) )
32		fveq1 5878	. . . . . 6 |- ( f = g -> ( f ` i ) = ( g ` i ) )
33	32	sseq1d 3445	. . . . 5 |- ( f = g -> ( ( f ` i ) C_ trCl ( X , A , R ) <-> ( g ` i ) C_ trCl ( X , A , R ) ) )
34	31, 33	imbi12d 327	. . . 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 4312	. . . . 5 |- ( i e. dom f -> ( f ` i ) C_ U_ i e. dom f ( f ` i ) )
36		ssiun2 4312	. . . . . 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 29809	. . . . . 6 |- trCl ( X , A , R ) = U_ f e. B U_ i e. dom f ( f ` i )
39	36, 38	syl6sseqr 3465	. . . . 5 |- ( f e. B -> U_ i e. dom f ( f ` i ) C_ trCl ( X , A , R ) )
40	35, 39	sylan9ssr 3432	. . . 4 |- ( ( f e. B /\ i e. dom f ) -> ( f ` i ) C_ trCl ( X , A , R ) )
41	27, 34, 40	chvar 2119	. . 3 |- ( ( g e. B /\ i e. dom g ) -> ( g ` i ) C_ trCl ( X , A , R ) )
42	21, 41	spei 2118	. 2 |- E. i ( ( g e. B /\ j e. dom g ) -> ( g ` j ) C_ trCl ( X , A , R ) )
43	15, 42	bnj1131 29671	1 |- ( ( g e. B /\ j e. dom g ) -> ( g ` j ) C_ trCl ( X , A , R ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   {cab 2457   A.wral 2756   E.wrex 2757   \ cdif 3387   C_ wss 3390   (/)c0 3722   {csn 3959   U_ciun 4269   dom cdm 4839   suc csuc 5432   Fn wfn 5584   ` cfv 5589   omcom 6711   predc-bnj14 29565   trClc-bnj18 29571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-dm 4849  df-iota 5553  df-fv 5597  df-bnj18 29572

This theorem is referenced by:  bnj1015  29844