Theorem bnj1280 31088

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
bnj1280.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1280.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1280.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1280.4	|- D = ( dom g i^i dom h )
bnj1280.5	|- E = { x e. D | ( g ` x ) =/= ( h ` x ) }
bnj1280.6	|- ( ph <-> ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
bnj1280.7	|- ( ps <-> ( ph /\ x e. E /\ A. y e. E -. y R x ) )
bnj1280.17	|- ( ps -> ( pred ( x , A , R ) i^i E ) = (/) )

Assertion

Ref	Expression
bnj1280	|- ( ps -> ( g |` pred ( x , A , R ) ) = ( h |` pred ( x , A , R ) ) )

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

Proof of Theorem bnj1280

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1280.1	. . . . . . . 8 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
2		bnj1280.2	. . . . . . . 8 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
3		bnj1280.3	. . . . . . . 8 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
4		bnj1280.4	. . . . . . . 8 |- D = ( dom g i^i dom h )
5		bnj1280.5	. . . . . . . 8 |- E = { x e. D | ( g ` x ) =/= ( h ` x ) }
6		bnj1280.6	. . . . . . . 8 |- ( ph <-> ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
7		bnj1280.7	. . . . . . . 8 |- ( ps <-> ( ph /\ x e. E /\ A. y e. E -. y R x ) )
8	1, 2, 3, 4, 5, 6, 7	bnj1286 31087	. . . . . . 7 |- ( ps -> pred ( x , A , R ) C_ D )
9	8	sseld 3602	. . . . . 6 |- ( ps -> ( z e. pred ( x , A , R ) -> z e. D ) )
10		bnj1280.17	. . . . . . . . 9 |- ( ps -> ( pred ( x , A , R ) i^i E ) = (/) )
11		disj1 4019	. . . . . . . . 9 |- ( ( pred ( x , A , R ) i^i E ) = (/) <-> A. z ( z e. pred ( x , A , R ) -> -. z e. E ) )
12	10, 11	sylib 208	. . . . . . . 8 |- ( ps -> A. z ( z e. pred ( x , A , R ) -> -. z e. E ) )
13	12	19.21bi 2059	. . . . . . 7 |- ( ps -> ( z e. pred ( x , A , R ) -> -. z e. E ) )
14		fveq2 6191	. . . . . . . . . . 11 |- ( x = z -> ( g ` x ) = ( g ` z ) )
15		fveq2 6191	. . . . . . . . . . 11 |- ( x = z -> ( h ` x ) = ( h ` z ) )
16	14, 15	neeq12d 2855	. . . . . . . . . 10 |- ( x = z -> ( ( g ` x ) =/= ( h ` x ) <-> ( g ` z ) =/= ( h ` z ) ) )
17	16, 5	elrab2 3366	. . . . . . . . 9 |- ( z e. E <-> ( z e. D /\ ( g ` z ) =/= ( h ` z ) ) )
18	17	notbii 310	. . . . . . . 8 |- ( -. z e. E <-> -. ( z e. D /\ ( g ` z ) =/= ( h ` z ) ) )
19		imnan 438	. . . . . . . 8 |- ( ( z e. D -> -. ( g ` z ) =/= ( h ` z ) ) <-> -. ( z e. D /\ ( g ` z ) =/= ( h ` z ) ) )
20		nne 2798	. . . . . . . . 9 |- ( -. ( g ` z ) =/= ( h ` z ) <-> ( g ` z ) = ( h ` z ) )
21	20	imbi2i 326	. . . . . . . 8 |- ( ( z e. D -> -. ( g ` z ) =/= ( h ` z ) ) <-> ( z e. D -> ( g ` z ) = ( h ` z ) ) )
22	18, 19, 21	3bitr2i 288	. . . . . . 7 |- ( -. z e. E <-> ( z e. D -> ( g ` z ) = ( h ` z ) ) )
23	13, 22	syl6ib 241	. . . . . 6 |- ( ps -> ( z e. pred ( x , A , R ) -> ( z e. D -> ( g ` z ) = ( h ` z ) ) ) )
24	9, 23	mpdd 43	. . . . 5 |- ( ps -> ( z e. pred ( x , A , R ) -> ( g ` z ) = ( h ` z ) ) )
25	24	imp 445	. . . 4 |- ( ( ps /\ z e. pred ( x , A , R ) ) -> ( g ` z ) = ( h ` z ) )
26		fvres 6207	. . . . . 6 |- ( z e. D -> ( ( g |` D ) ` z ) = ( g ` z ) )
27	9, 26	syl6 35	. . . . 5 |- ( ps -> ( z e. pred ( x , A , R ) -> ( ( g |` D ) ` z ) = ( g ` z ) ) )
28	27	imp 445	. . . 4 |- ( ( ps /\ z e. pred ( x , A , R ) ) -> ( ( g |` D ) ` z ) = ( g ` z ) )
29		fvres 6207	. . . . . 6 |- ( z e. D -> ( ( h |` D ) ` z ) = ( h ` z ) )
30	9, 29	syl6 35	. . . . 5 |- ( ps -> ( z e. pred ( x , A , R ) -> ( ( h |` D ) ` z ) = ( h ` z ) ) )
31	30	imp 445	. . . 4 |- ( ( ps /\ z e. pred ( x , A , R ) ) -> ( ( h |` D ) ` z ) = ( h ` z ) )
32	25, 28, 31	3eqtr4d 2666	. . 3 |- ( ( ps /\ z e. pred ( x , A , R ) ) -> ( ( g |` D ) ` z ) = ( ( h |` D ) ` z ) )
33	32	ralrimiva 2966	. 2 |- ( ps -> A. z e. pred ( x , A , R ) ( ( g |` D ) ` z ) = ( ( h |` D ) ` z ) )
34	8	resabs1d 5428	. . . 4 |- ( ps -> ( ( g |` D ) |` pred ( x , A , R ) ) = ( g |` pred ( x , A , R ) ) )
35	8	resabs1d 5428	. . . 4 |- ( ps -> ( ( h |` D ) |` pred ( x , A , R ) ) = ( h |` pred ( x , A , R ) ) )
36	34, 35	eqeq12d 2637	. . 3 |- ( ps -> ( ( ( g |` D ) |` pred ( x , A , R ) ) = ( ( h |` D ) |` pred ( x , A , R ) ) <-> ( g |` pred ( x , A , R ) ) = ( h |` pred ( x , A , R ) ) ) )
37	1, 2, 3, 4, 5, 6, 7	bnj1256 31083	. . . . . . 7 |- ( ph -> E. d e. B g Fn d )
38	4	bnj1292 30886	. . . . . . . . 9 |- D C_ dom g
39		fndm 5990	. . . . . . . . 9 |- ( g Fn d -> dom g = d )
40	38, 39	syl5sseq 3653	. . . . . . . 8 |- ( g Fn d -> D C_ d )
41		fnssres 6004	. . . . . . . 8 |- ( ( g Fn d /\ D C_ d ) -> ( g |` D ) Fn D )
42	40, 41	mpdan 702	. . . . . . 7 |- ( g Fn d -> ( g |` D ) Fn D )
43	37, 42	bnj31 30785	. . . . . 6 |- ( ph -> E. d e. B ( g |` D ) Fn D )
44	43	bnj1265 30883	. . . . 5 |- ( ph -> ( g |` D ) Fn D )
45	7, 44	bnj835 30829	. . . 4 |- ( ps -> ( g |` D ) Fn D )
46	1, 2, 3, 4, 5, 6, 7	bnj1259 31084	. . . . . . 7 |- ( ph -> E. d e. B h Fn d )
47	4	bnj1293 30887	. . . . . . . . 9 |- D C_ dom h
48		fndm 5990	. . . . . . . . 9 |- ( h Fn d -> dom h = d )
49	47, 48	syl5sseq 3653	. . . . . . . 8 |- ( h Fn d -> D C_ d )
50		fnssres 6004	. . . . . . . 8 |- ( ( h Fn d /\ D C_ d ) -> ( h |` D ) Fn D )
51	49, 50	mpdan 702	. . . . . . 7 |- ( h Fn d -> ( h |` D ) Fn D )
52	46, 51	bnj31 30785	. . . . . 6 |- ( ph -> E. d e. B ( h |` D ) Fn D )
53	52	bnj1265 30883	. . . . 5 |- ( ph -> ( h |` D ) Fn D )
54	7, 53	bnj835 30829	. . . 4 |- ( ps -> ( h |` D ) Fn D )
55		fvreseq 6319	. . . 4 |- ( ( ( ( g |` D ) Fn D /\ ( h |` D ) Fn D ) /\ pred ( x , A , R ) C_ D ) -> ( ( ( g |` D ) |` pred ( x , A , R ) ) = ( ( h |` D ) |` pred ( x , A , R ) ) <-> A. z e. pred ( x , A , R ) ( ( g |` D ) ` z ) = ( ( h |` D ) ` z ) ) )
56	45, 54, 8, 55	syl21anc 1325	. . 3 |- ( ps -> ( ( ( g |` D ) |` pred ( x , A , R ) ) = ( ( h |` D ) |` pred ( x , A , R ) ) <-> A. z e. pred ( x , A , R ) ( ( g |` D ) ` z ) = ( ( h |` D ) ` z ) ) )
57	36, 56	bitr3d 270	. 2 |- ( ps -> ( ( g |` pred ( x , A , R ) ) = ( h |` pred ( x , A , R ) ) <-> A. z e. pred ( x , A , R ) ( ( g |` D ) ` z ) = ( ( h |` D ) ` z ) ) )
58	33, 57	mpbird 247	1 |- ( ps -> ( g |` pred ( x , A , R ) ) = ( h |` pred ( x , A , R ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   (/)c0 3915   <.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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-bnj17 30753

This theorem is referenced by:  bnj1311  31092