Theorem bnj1286 31087

Description: Technical lemma for bnj60 31130. 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
bnj1286.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1286.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1286.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1286.4	|- D = ( dom g i^i dom h )
bnj1286.5	|- E = { x e. D | ( g ` x ) =/= ( h ` x ) }
bnj1286.6	|- ( ph <-> ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
bnj1286.7	|- ( ps <-> ( ph /\ x e. E /\ A. y e. E -. y R x ) )

Assertion

Ref	Expression
bnj1286	|- ( ps -> pred ( x , A , R ) C_ D )

Distinct variable groups:   A, d, f   B, f, g   B, h, f   x, D   f, G, g   h, G   R, d, f   g, Y   h, Y   g, d, x, f   h, d, x

Allowed substitution hints:   ph( x, y, f, g, h, d)   ps( x, y, f, g, h, d)   A( x, y, g, h)   B( x, y, d)   C( x, y, f, g, h, d)   D( y, f, g, h, d)   R( x, y, g, h)   E( x, y, f, g, h, d)   G( x, y, d)   Y( x, y, f, d)

Proof of Theorem bnj1286

Step	Hyp	Ref	Expression
1		bnj1286.7	. . . . 5 |- ( ps <-> ( ph /\ x e. E /\ A. y e. E -. y R x ) )
2		bnj1286.1	. . . . . . . . 9 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
3		bnj1286.2	. . . . . . . . 9 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
4		bnj1286.3	. . . . . . . . 9 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
5		bnj1286.4	. . . . . . . . 9 |- D = ( dom g i^i dom h )
6		bnj1286.5	. . . . . . . . 9 |- E = { x e. D | ( g ` x ) =/= ( h ` x ) }
7		bnj1286.6	. . . . . . . . 9 |- ( ph <-> ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
8	2, 3, 4, 5, 6, 7, 1	bnj1256 31083	. . . . . . . 8 |- ( ph -> E. d e. B g Fn d )
9	8	bnj1196 30865	. . . . . . 7 |- ( ph -> E. d ( d e. B /\ g Fn d ) )
10	2	bnj1517 30920	. . . . . . . . 9 |- ( d e. B -> A. x e. d pred ( x , A , R ) C_ d )
11	10	adantr 481	. . . . . . . 8 |- ( ( d e. B /\ g Fn d ) -> A. x e. d pred ( x , A , R ) C_ d )
12		fndm 5990	. . . . . . . . . 10 |- ( g Fn d -> dom g = d )
13		sseq2 3627	. . . . . . . . . . 11 |- ( dom g = d -> ( pred ( x , A , R ) C_ dom g <-> pred ( x , A , R ) C_ d ) )
14	13	raleqbi1dv 3146	. . . . . . . . . 10 |- ( dom g = d -> ( A. x e. dom g pred ( x , A , R ) C_ dom g <-> A. x e. d pred ( x , A , R ) C_ d ) )
15	12, 14	syl 17	. . . . . . . . 9 |- ( g Fn d -> ( A. x e. dom g pred ( x , A , R ) C_ dom g <-> A. x e. d pred ( x , A , R ) C_ d ) )
16	15	adantl 482	. . . . . . . 8 |- ( ( d e. B /\ g Fn d ) -> ( A. x e. dom g pred ( x , A , R ) C_ dom g <-> A. x e. d pred ( x , A , R ) C_ d ) )
17	11, 16	mpbird 247	. . . . . . 7 |- ( ( d e. B /\ g Fn d ) -> A. x e. dom g pred ( x , A , R ) C_ dom g )
18	9, 17	bnj593 30815	. . . . . 6 |- ( ph -> E. d A. x e. dom g pred ( x , A , R ) C_ dom g )
19	18	bnj937 30842	. . . . 5 |- ( ph -> A. x e. dom g pred ( x , A , R ) C_ dom g )
20	1, 19	bnj835 30829	. . . 4 |- ( ps -> A. x e. dom g pred ( x , A , R ) C_ dom g )
21	6	ssrab3 3688	. . . . . . 7 |- E C_ D
22	5	bnj1292 30886	. . . . . . 7 |- D C_ dom g
23	21, 22	sstri 3612	. . . . . 6 |- E C_ dom g
24	23	sseli 3599	. . . . 5 |- ( x e. E -> x e. dom g )
25	1, 24	bnj836 30830	. . . 4 |- ( ps -> x e. dom g )
26	20, 25	bnj1294 30888	. . 3 |- ( ps -> pred ( x , A , R ) C_ dom g )
27	2, 3, 4, 5, 6, 7, 1	bnj1259 31084	. . . . . . . 8 |- ( ph -> E. d e. B h Fn d )
28	27	bnj1196 30865	. . . . . . 7 |- ( ph -> E. d ( d e. B /\ h Fn d ) )
29	10	adantr 481	. . . . . . . 8 |- ( ( d e. B /\ h Fn d ) -> A. x e. d pred ( x , A , R ) C_ d )
30		fndm 5990	. . . . . . . . . 10 |- ( h Fn d -> dom h = d )
31		sseq2 3627	. . . . . . . . . . 11 |- ( dom h = d -> ( pred ( x , A , R ) C_ dom h <-> pred ( x , A , R ) C_ d ) )
32	31	raleqbi1dv 3146	. . . . . . . . . 10 |- ( dom h = d -> ( A. x e. dom h pred ( x , A , R ) C_ dom h <-> A. x e. d pred ( x , A , R ) C_ d ) )
33	30, 32	syl 17	. . . . . . . . 9 |- ( h Fn d -> ( A. x e. dom h pred ( x , A , R ) C_ dom h <-> A. x e. d pred ( x , A , R ) C_ d ) )
34	33	adantl 482	. . . . . . . 8 |- ( ( d e. B /\ h Fn d ) -> ( A. x e. dom h pred ( x , A , R ) C_ dom h <-> A. x e. d pred ( x , A , R ) C_ d ) )
35	29, 34	mpbird 247	. . . . . . 7 |- ( ( d e. B /\ h Fn d ) -> A. x e. dom h pred ( x , A , R ) C_ dom h )
36	28, 35	bnj593 30815	. . . . . 6 |- ( ph -> E. d A. x e. dom h pred ( x , A , R ) C_ dom h )
37	36	bnj937 30842	. . . . 5 |- ( ph -> A. x e. dom h pred ( x , A , R ) C_ dom h )
38	1, 37	bnj835 30829	. . . 4 |- ( ps -> A. x e. dom h pred ( x , A , R ) C_ dom h )
39	5	bnj1293 30887	. . . . . . 7 |- D C_ dom h
40	21, 39	sstri 3612	. . . . . 6 |- E C_ dom h
41	40	sseli 3599	. . . . 5 |- ( x e. E -> x e. dom h )
42	1, 41	bnj836 30830	. . . 4 |- ( ps -> x e. dom h )
43	38, 42	bnj1294 30888	. . 3 |- ( ps -> pred ( x , A , R ) C_ dom h )
44	26, 43	ssind 3837	. 2 |- ( ps -> pred ( x , A , R ) C_ ( dom g i^i dom h ) )
45	44, 5	syl6sseqr 3652	1 |- ( ps -> pred ( x , A , R ) C_ D )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   <.cop 4183   class class class wbr 4653   dom cdm 5114   |` cres 5116   Fn wfn 5883   ` cfv 5888   /\ w-bnj17 30752   predc-bnj14 30754   FrSe w-bnj15 30758

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-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-bnj17 30753

This theorem is referenced by:  bnj1280  31088