Theorem bnj1015 31031

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
bnj1015.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj1015.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj1015.13	|- D = ( om \ { (/) } )
bnj1015.14	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj1015.15	|- G e. V
bnj1015.16	|- J e. V

Assertion

Ref	Expression
bnj1015	|- ( ( 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   ph, i

Allowed substitution hints:   ph( y, f, n)   ps( y, f, i, n)   B( y, f, i, n)   D( y, f, n)   G( y, f, i, n)   J( y, f, i, n)   V( y, f, i, n)

Proof of Theorem bnj1015

Dummy variables g j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1015.16	. . 3 |- J e. V
2	1	elexi 3213	. 2 |- J e. _V
3		eleq1 2689	. . . 4 |- ( j = J -> ( j e. dom G <-> J e. dom G ) )
4	3	anbi2d 740	. . 3 |- ( j = J -> ( ( G e. B /\ j e. dom G ) <-> ( G e. B /\ J e. dom G ) ) )
5		fveq2 6191	. . . 4 |- ( j = J -> ( G ` j ) = ( G ` J ) )
6	5	sseq1d 3632	. . 3 |- ( j = J -> ( ( G ` j ) C_ trCl ( X , A , R ) <-> ( G ` J ) C_ trCl ( X , A , R ) ) )
7	4, 6	imbi12d 334	. 2 |- ( j = J -> ( ( ( G e. B /\ j e. dom G ) -> ( G ` j ) C_ trCl ( X , A , R ) ) <-> ( ( G e. B /\ J e. dom G ) -> ( G ` J ) C_ trCl ( X , A , R ) ) ) )
8		bnj1015.15	. . . 4 |- G e. V
9	8	elexi 3213	. . 3 |- G e. _V
10		eleq1 2689	. . . . 5 |- ( g = G -> ( g e. B <-> G e. B ) )
11		dmeq 5324	. . . . . 6 |- ( g = G -> dom g = dom G )
12	11	eleq2d 2687	. . . . 5 |- ( g = G -> ( j e. dom g <-> j e. dom G ) )
13	10, 12	anbi12d 747	. . . 4 |- ( g = G -> ( ( g e. B /\ j e. dom g ) <-> ( G e. B /\ j e. dom G ) ) )
14		fveq1 6190	. . . . 5 |- ( g = G -> ( g ` j ) = ( G ` j ) )
15	14	sseq1d 3632	. . . 4 |- ( g = G -> ( ( g ` j ) C_ trCl ( X , A , R ) <-> ( G ` j ) C_ trCl ( X , A , R ) ) )
16	13, 15	imbi12d 334	. . 3 |- ( g = G -> ( ( ( g e. B /\ j e. dom g ) -> ( g ` j ) C_ trCl ( X , A , R ) ) <-> ( ( G e. B /\ j e. dom G ) -> ( G ` j ) C_ trCl ( X , A , R ) ) ) )
17		bnj1015.1	. . . 4 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
18		bnj1015.2	. . . 4 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
19		bnj1015.13	. . . 4 |- D = ( om \ { (/) } )
20		bnj1015.14	. . . 4 |- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
21	17, 18, 19, 20	bnj1014 31030	. . 3 |- ( ( g e. B /\ j e. dom g ) -> ( g ` j ) C_ trCl ( X , A , R ) )
22	9, 16, 21	vtocl 3259	. 2 |- ( ( G e. B /\ j e. dom G ) -> ( G ` j ) C_ trCl ( X , A , R ) )
23	2, 7, 22	vtocl 3259	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:  bnj1018  31032