Theorem bnj1014 31030

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
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 2764	. . . . . . . . 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 31002	. . . . . . . . . 10 |- ( ( f Fn n /\ ph /\ ps ) -> A. i ( f Fn n /\ ph /\ ps ) )
6	5	nf5i 2024	. . . . . . . . 9 |- F/ i ( f Fn n /\ ph /\ ps )
7	2, 6	nfrex 3007	. . . . . . . 8 |- F/ i E. n e. D ( f Fn n /\ ph /\ ps )
8	7	nfab 2769	. . . . . . 7 |- F/_ i { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
9	1, 8	nfcxfr 2762	. . . . . 6 |- F/_ i B
10	9	nfcri 2758	. . . . 5 |- F/ i g e. B
11		nfv 1843	. . . . 5 |- F/ i j e. dom g
12	10, 11	nfan 1828	. . . 4 |- F/ i ( g e. B /\ j e. dom g )
13		nfv 1843	. . . 4 |- F/ i ( g ` j ) C_ trCl ( X , A , R )
14	12, 13	nfim 1825	. . 3 |- F/ i ( ( g e. B /\ j e. dom g ) -> ( g ` j ) C_ trCl ( X , A , R ) )
15	14	nf5ri 2065	. 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 2689	. . . . . 6 |- ( j = i -> ( j e. dom g <-> i e. dom g ) )
17	16	anbi2d 740	. . . . 5 |- ( j = i -> ( ( g e. B /\ j e. dom g ) <-> ( g e. B /\ i e. dom g ) ) )
18		fveq2 6191	. . . . . 6 |- ( j = i -> ( g ` j ) = ( g ` i ) )
19	18	sseq1d 3632	. . . . 5 |- ( j = i -> ( ( g ` j ) C_ trCl ( X , A , R ) <-> ( g ` i ) C_ trCl ( X , A , R ) ) )
20	17, 19	imbi12d 334	. . . 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 1947	. . 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 30892	. . . . . . 7 |- ( g e. B -> A. f g e. B )
23	22	nf5i 2024	. . . . . 6 |- F/ f g e. B
24		nfv 1843	. . . . . 6 |- F/ f i e. dom g
25	23, 24	nfan 1828	. . . . 5 |- F/ f ( g e. B /\ i e. dom g )
26		nfv 1843	. . . . 5 |- F/ f ( g ` i ) C_ trCl ( X , A , R )
27	25, 26	nfim 1825	. . . 4 |- F/ f ( ( g e. B /\ i e. dom g ) -> ( g ` i ) C_ trCl ( X , A , R ) )
28		eleq1 2689	. . . . . 6 |- ( f = g -> ( f e. B <-> g e. B ) )
29		dmeq 5324	. . . . . . 7 |- ( f = g -> dom f = dom g )
30	29	eleq2d 2687	. . . . . 6 |- ( f = g -> ( i e. dom f <-> i e. dom g ) )
31	28, 30	anbi12d 747	. . . . 5 |- ( f = g -> ( ( f e. B /\ i e. dom f ) <-> ( g e. B /\ i e. dom g ) ) )
32		fveq1 6190	. . . . . 6 |- ( f = g -> ( f ` i ) = ( g ` i ) )
33	32	sseq1d 3632	. . . . 5 |- ( f = g -> ( ( f ` i ) C_ trCl ( X , A , R ) <-> ( g ` i ) C_ trCl ( X , A , R ) ) )
34	31, 33	imbi12d 334	. . . 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 4563	. . . . 5 |- ( i e. dom f -> ( f ` i ) C_ U_ i e. dom f ( f ` i ) )
36		ssiun2 4563	. . . . . 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 30996	. . . . . 6 |- trCl ( X , A , R ) = U_ f e. B U_ i e. dom f ( f ` i )
39	36, 38	syl6sseqr 3652	. . . . 5 |- ( f e. B -> U_ i e. dom f ( f ` i ) C_ trCl ( X , A , R ) )
40	35, 39	sylan9ssr 3617	. . . 4 |- ( ( f e. B /\ i e. dom f ) -> ( f ` i ) C_ trCl ( X , A , R ) )
41	27, 34, 40	chvar 2262	. . 3 |- ( ( g e. B /\ i e. dom g ) -> ( g ` i ) C_ trCl ( X , A , R ) )
42	21, 41	spei 2261	. 2 |- E. i ( ( g e. B /\ j e. dom g ) -> ( g ` j ) C_ trCl ( X , A , R ) )
43	15, 42	bnj1131 30858	1 |- ( ( g e. B /\ j e. dom g ) -> ( g ` j ) C_ trCl ( X , A , R ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   U_ciun 4520   dom cdm 5114   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   predc-bnj14 30754   trClc-bnj18 30760

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-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-dm 5124  df-iota 5851  df-fv 5896  df-bnj18 30761

This theorem is referenced by:  bnj1015  31031