Theorem bnj1123 31054

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
bnj1123.4	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj1123.3	|- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj1123.1	|- ( et <-> ( ( f e. K /\ i e. dom f ) -> ( f ` i ) C_ B ) )
bnj1123.2	|- ( et' <-> [. j / i ]. et )

Assertion

Ref	Expression
bnj1123	|- ( et' <-> ( ( f e. K /\ j e. dom f ) -> ( f ` j ) C_ B ) )

Distinct variable groups:   B, i   D, i   f, i   i, j   i, n   ph, i

Allowed substitution hints:   ph( y, f, j, n)   ps( y, f, i, j, n)   et( y, f, i, j, n)   A( y, f, i, j, n)   B( y, f, j, n)   D( y, f, j, n)   R( y, f, i, j, n)   K( y, f, i, j, n)   et'( y, f, i, j, n)

Proof of Theorem bnj1123

Step	Hyp	Ref	Expression
1		bnj1123.2	. 2 |- ( et' <-> [. j / i ]. et )
2		bnj1123.1	. . 3 |- ( et <-> ( ( f e. K /\ i e. dom f ) -> ( f ` i ) C_ B ) )
3	2	sbcbii 3491	. 2 |- ( [. j / i ]. et <-> [. j / i ]. ( ( f e. K /\ i e. dom f ) -> ( f ` i ) C_ B ) )
4		vex 3203	. . 3 |- j e. _V
5		bnj1123.3	. . . . . . . 8 |- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
6		nfcv 2764	. . . . . . . . . 10 |- F/_ i D
7		nfv 1843	. . . . . . . . . . 11 |- F/ i f Fn n
8		nfv 1843	. . . . . . . . . . 11 |- F/ i ph
9		bnj1123.4	. . . . . . . . . . . . 13 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
10	9	bnj1095 30852	. . . . . . . . . . . 12 |- ( ps -> A. i ps )
11	10	nf5i 2024	. . . . . . . . . . 11 |- F/ i ps
12	7, 8, 11	nf3an 1831	. . . . . . . . . 10 |- F/ i ( f Fn n /\ ph /\ ps )
13	6, 12	nfrex 3007	. . . . . . . . 9 |- F/ i E. n e. D ( f Fn n /\ ph /\ ps )
14	13	nfab 2769	. . . . . . . 8 |- F/_ i { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
15	5, 14	nfcxfr 2762	. . . . . . 7 |- F/_ i K
16	15	nfcri 2758	. . . . . 6 |- F/ i f e. K
17		nfv 1843	. . . . . 6 |- F/ i j e. dom f
18	16, 17	nfan 1828	. . . . 5 |- F/ i ( f e. K /\ j e. dom f )
19		nfv 1843	. . . . 5 |- F/ i ( f ` j ) C_ B
20	18, 19	nfim 1825	. . . 4 |- F/ i ( ( f e. K /\ j e. dom f ) -> ( f ` j ) C_ B )
21		eleq1 2689	. . . . . 6 |- ( i = j -> ( i e. dom f <-> j e. dom f ) )
22	21	anbi2d 740	. . . . 5 |- ( i = j -> ( ( f e. K /\ i e. dom f ) <-> ( f e. K /\ j e. dom f ) ) )
23		fveq2 6191	. . . . . 6 |- ( i = j -> ( f ` i ) = ( f ` j ) )
24	23	sseq1d 3632	. . . . 5 |- ( i = j -> ( ( f ` i ) C_ B <-> ( f ` j ) C_ B ) )
25	22, 24	imbi12d 334	. . . 4 |- ( i = j -> ( ( ( f e. K /\ i e. dom f ) -> ( f ` i ) C_ B ) <-> ( ( f e. K /\ j e. dom f ) -> ( f ` j ) C_ B ) ) )
26	20, 25	sbciegf 3467	. . 3 |- ( j e. _V -> ( [. j / i ]. ( ( f e. K /\ i e. dom f ) -> ( f ` i ) C_ B ) <-> ( ( f e. K /\ j e. dom f ) -> ( f ` j ) C_ B ) ) )
27	4, 26	ax-mp 5	. 2 |- ( [. j / i ]. ( ( f e. K /\ i e. dom f ) -> ( f ` i ) C_ B ) <-> ( ( f e. K /\ j e. dom f ) -> ( f ` j ) C_ B ) )
28	1, 3, 27	3bitri 286	1 |- ( et' <-> ( ( f e. K /\ j e. dom f ) -> ( f ` j ) C_ B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   [.wsbc 3435   C_ wss 3574   U_ciun 4520   dom cdm 5114   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   predc-bnj14 30754

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-sbc 3436  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-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj1030  31055