Theorem bnj1253 31085

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
bnj1253.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1253.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1253.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1253.4	|- D = ( dom g i^i dom h )
bnj1253.5	|- E = { x e. D | ( g ` x ) =/= ( h ` x ) }
bnj1253.6	|- ( ph <-> ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
bnj1253.7	|- ( ps <-> ( ph /\ x e. E /\ A. y e. E -. y R x ) )

Assertion

Ref	Expression
bnj1253	|- ( ph -> E =/= (/) )

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

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

Proof of Theorem bnj1253

Step	Hyp	Ref	Expression
1		bnj1253.6	. . . 4 |- ( ph <-> ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
2	1	bnj1254 30880	. . 3 |- ( ph -> ( g |` D ) =/= ( h |` D ) )
3		bnj1253.1	. . . . . . . . . . 11 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
4		bnj1253.2	. . . . . . . . . . 11 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
5		bnj1253.3	. . . . . . . . . . 11 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
6		bnj1253.4	. . . . . . . . . . 11 |- D = ( dom g i^i dom h )
7		bnj1253.5	. . . . . . . . . . 11 |- E = { x e. D | ( g ` x ) =/= ( h ` x ) }
8		bnj1253.7	. . . . . . . . . . 11 |- ( ps <-> ( ph /\ x e. E /\ A. y e. E -. y R x ) )
9	3, 4, 5, 6, 7, 1, 8	bnj1256 31083	. . . . . . . . . 10 |- ( ph -> E. d e. B g Fn d )
10	6	bnj1292 30886	. . . . . . . . . . . 12 |- D C_ dom g
11		fndm 5990	. . . . . . . . . . . 12 |- ( g Fn d -> dom g = d )
12	10, 11	syl5sseq 3653	. . . . . . . . . . 11 |- ( g Fn d -> D C_ d )
13		fnssres 6004	. . . . . . . . . . 11 |- ( ( g Fn d /\ D C_ d ) -> ( g |` D ) Fn D )
14	12, 13	mpdan 702	. . . . . . . . . 10 |- ( g Fn d -> ( g |` D ) Fn D )
15	9, 14	bnj31 30785	. . . . . . . . 9 |- ( ph -> E. d e. B ( g |` D ) Fn D )
16	15	bnj1265 30883	. . . . . . . 8 |- ( ph -> ( g |` D ) Fn D )
17	3, 4, 5, 6, 7, 1, 8	bnj1259 31084	. . . . . . . . . 10 |- ( ph -> E. d e. B h Fn d )
18	6	bnj1293 30887	. . . . . . . . . . . 12 |- D C_ dom h
19		fndm 5990	. . . . . . . . . . . 12 |- ( h Fn d -> dom h = d )
20	18, 19	syl5sseq 3653	. . . . . . . . . . 11 |- ( h Fn d -> D C_ d )
21		fnssres 6004	. . . . . . . . . . 11 |- ( ( h Fn d /\ D C_ d ) -> ( h |` D ) Fn D )
22	20, 21	mpdan 702	. . . . . . . . . 10 |- ( h Fn d -> ( h |` D ) Fn D )
23	17, 22	bnj31 30785	. . . . . . . . 9 |- ( ph -> E. d e. B ( h |` D ) Fn D )
24	23	bnj1265 30883	. . . . . . . 8 |- ( ph -> ( h |` D ) Fn D )
25		ssid 3624	. . . . . . . . 9 |- D C_ D
26		fvreseq 6319	. . . . . . . . 9 |- ( ( ( ( g |` D ) Fn D /\ ( h |` D ) Fn D ) /\ D C_ D ) -> ( ( ( g |` D ) |` D ) = ( ( h |` D ) |` D ) <-> A. x e. D ( ( g |` D ) ` x ) = ( ( h |` D ) ` x ) ) )
27	25, 26	mpan2 707	. . . . . . . 8 |- ( ( ( g |` D ) Fn D /\ ( h |` D ) Fn D ) -> ( ( ( g |` D ) |` D ) = ( ( h |` D ) |` D ) <-> A. x e. D ( ( g |` D ) ` x ) = ( ( h |` D ) ` x ) ) )
28	16, 24, 27	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( ( g |` D ) |` D ) = ( ( h |` D ) |` D ) <-> A. x e. D ( ( g |` D ) ` x ) = ( ( h |` D ) ` x ) ) )
29		residm 5430	. . . . . . . 8 |- ( ( g |` D ) |` D ) = ( g |` D )
30		residm 5430	. . . . . . . 8 |- ( ( h |` D ) |` D ) = ( h |` D )
31	29, 30	eqeq12i 2636	. . . . . . 7 |- ( ( ( g |` D ) |` D ) = ( ( h |` D ) |` D ) <-> ( g |` D ) = ( h |` D ) )
32		df-ral 2917	. . . . . . 7 |- ( A. x e. D ( ( g |` D ) ` x ) = ( ( h |` D ) ` x ) <-> A. x ( x e. D -> ( ( g |` D ) ` x ) = ( ( h |` D ) ` x ) ) )
33	28, 31, 32	3bitr3g 302	. . . . . 6 |- ( ph -> ( ( g |` D ) = ( h |` D ) <-> A. x ( x e. D -> ( ( g |` D ) ` x ) = ( ( h |` D ) ` x ) ) ) )
34		fvres 6207	. . . . . . . . 9 |- ( x e. D -> ( ( g |` D ) ` x ) = ( g ` x ) )
35		fvres 6207	. . . . . . . . 9 |- ( x e. D -> ( ( h |` D ) ` x ) = ( h ` x ) )
36	34, 35	eqeq12d 2637	. . . . . . . 8 |- ( x e. D -> ( ( ( g |` D ) ` x ) = ( ( h |` D ) ` x ) <-> ( g ` x ) = ( h ` x ) ) )
37	36	pm5.74i 260	. . . . . . 7 |- ( ( x e. D -> ( ( g |` D ) ` x ) = ( ( h |` D ) ` x ) ) <-> ( x e. D -> ( g ` x ) = ( h ` x ) ) )
38	37	albii 1747	. . . . . 6 |- ( A. x ( x e. D -> ( ( g |` D ) ` x ) = ( ( h |` D ) ` x ) ) <-> A. x ( x e. D -> ( g ` x ) = ( h ` x ) ) )
39	33, 38	syl6bb 276	. . . . 5 |- ( ph -> ( ( g |` D ) = ( h |` D ) <-> A. x ( x e. D -> ( g ` x ) = ( h ` x ) ) ) )
40	39	necon3abid 2830	. . . 4 |- ( ph -> ( ( g |` D ) =/= ( h |` D ) <-> -. A. x ( x e. D -> ( g ` x ) = ( h ` x ) ) ) )
41		df-rex 2918	. . . . 5 |- ( E. x e. D ( g ` x ) =/= ( h ` x ) <-> E. x ( x e. D /\ ( g ` x ) =/= ( h ` x ) ) )
42		pm4.61 442	. . . . . . 7 |- ( -. ( x e. D -> ( g ` x ) = ( h ` x ) ) <-> ( x e. D /\ -. ( g ` x ) = ( h ` x ) ) )
43		df-ne 2795	. . . . . . . 8 |- ( ( g ` x ) =/= ( h ` x ) <-> -. ( g ` x ) = ( h ` x ) )
44	43	anbi2i 730	. . . . . . 7 |- ( ( x e. D /\ ( g ` x ) =/= ( h ` x ) ) <-> ( x e. D /\ -. ( g ` x ) = ( h ` x ) ) )
45	42, 44	bitr4i 267	. . . . . 6 |- ( -. ( x e. D -> ( g ` x ) = ( h ` x ) ) <-> ( x e. D /\ ( g ` x ) =/= ( h ` x ) ) )
46	45	exbii 1774	. . . . 5 |- ( E. x -. ( x e. D -> ( g ` x ) = ( h ` x ) ) <-> E. x ( x e. D /\ ( g ` x ) =/= ( h ` x ) ) )
47		exnal 1754	. . . . 5 |- ( E. x -. ( x e. D -> ( g ` x ) = ( h ` x ) ) <-> -. A. x ( x e. D -> ( g ` x ) = ( h ` x ) ) )
48	41, 46, 47	3bitr2ri 289	. . . 4 |- ( -. A. x ( x e. D -> ( g ` x ) = ( h ` x ) ) <-> E. x e. D ( g ` x ) =/= ( h ` x ) )
49	40, 48	syl6bb 276	. . 3 |- ( ph -> ( ( g |` D ) =/= ( h |` D ) <-> E. x e. D ( g ` x ) =/= ( h ` x ) ) )
50	2, 49	mpbid 222	. 2 |- ( ph -> E. x e. D ( g ` x ) =/= ( h ` x ) )
51	7	neeq1i 2858	. . 3 |- ( E =/= (/) <-> { x e. D | ( g ` x ) =/= ( h ` x ) } =/= (/) )
52		rabn0 3958	. . 3 |- ( { x e. D | ( g ` x ) =/= ( h ` x ) } =/= (/) <-> E. x e. D ( g ` x ) =/= ( h ` x ) )
53	51, 52	bitri 264	. 2 |- ( E =/= (/) <-> E. x e. D ( g ` x ) =/= ( h ` x ) )
54	50, 53	sylibr 224	1 |- ( ph -> E =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   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