Theorem bnj1311 31092

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
bnj1311.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1311.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1311.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1311.4	|- D = ( dom g i^i dom h )

Assertion

Ref	Expression
bnj1311	|- ( ( R FrSe A /\ g e. C /\ h e. C ) -> ( g |` D ) = ( h |` D ) )

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

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

Proof of Theorem bnj1311

Dummy variables w z y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		biid 251	. . . . . . . 8 |- ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) <-> ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
2	1	bnj1232 30874	. . . . . . 7 |- ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) -> R FrSe A )
3		ssrab2 3687	. . . . . . . 8 |- { x e. D | ( g ` x ) =/= ( h ` x ) } C_ D
4		bnj1311.4	. . . . . . . . 9 |- D = ( dom g i^i dom h )
5	1	bnj1235 30875	. . . . . . . . . . 11 |- ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) -> g e. C )
6		bnj1311.2	. . . . . . . . . . . 12 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
7		bnj1311.3	. . . . . . . . . . . 12 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
8		eqid 2622	. . . . . . . . . . . 12 |- <. x , ( g |` pred ( x , A , R ) ) >. = <. x , ( g |` pred ( x , A , R ) ) >.
9		eqid 2622	. . . . . . . . . . . 12 |- { g | E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) ) } = { g | E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) ) }
10	6, 7, 8, 9	bnj1234 31081	. . . . . . . . . . 11 |- C = { g | E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) ) }
11	5, 10	syl6eleq 2711	. . . . . . . . . 10 |- ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) -> g e. { g | E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) ) } )
12		abid 2610	. . . . . . . . . . . . . 14 |- ( g e. { g | E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) ) } <-> E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) ) )
13	12	bnj1238 30877	. . . . . . . . . . . . 13 |- ( g e. { g | E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) ) } -> E. d e. B g Fn d )
14	13	bnj1196 30865	. . . . . . . . . . . 12 |- ( g e. { g | E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) ) } -> E. d ( d e. B /\ g Fn d ) )
15		bnj1311.1	. . . . . . . . . . . . . . 15 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
16	15	abeq2i 2735	. . . . . . . . . . . . . 14 |- ( d e. B <-> ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) )
17	16	simplbi 476	. . . . . . . . . . . . 13 |- ( d e. B -> d C_ A )
18		fndm 5990	. . . . . . . . . . . . 13 |- ( g Fn d -> dom g = d )
19	17, 18	bnj1241 30878	. . . . . . . . . . . 12 |- ( ( d e. B /\ g Fn d ) -> dom g C_ A )
20	14, 19	bnj593 30815	. . . . . . . . . . 11 |- ( g e. { g | E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) ) } -> E. d dom g C_ A )
21	20	bnj937 30842	. . . . . . . . . 10 |- ( g e. { g | E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) ) } -> dom g C_ A )
22		ssinss1 3841	. . . . . . . . . 10 |- ( dom g C_ A -> ( dom g i^i dom h ) C_ A )
23	11, 21, 22	3syl 18	. . . . . . . . 9 |- ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) -> ( dom g i^i dom h ) C_ A )
24	4, 23	syl5eqss 3649	. . . . . . . 8 |- ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) -> D C_ A )
25	3, 24	syl5ss 3614	. . . . . . 7 |- ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) -> { x e. D | ( g ` x ) =/= ( h ` x ) } C_ A )
26		eqid 2622	. . . . . . . 8 |- { x e. D | ( g ` x ) =/= ( h ` x ) } = { x e. D | ( g ` x ) =/= ( h ` x ) }
27		biid 251	. . . . . . . 8 |- ( ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) /\ x e. { x e. D | ( g ` x ) =/= ( h ` x ) } /\ A. y e. { x e. D | ( g ` x ) =/= ( h ` x ) } -. y R x ) <-> ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) /\ x e. { x e. D | ( g ` x ) =/= ( h ` x ) } /\ A. y e. { x e. D | ( g ` x ) =/= ( h ` x ) } -. y R x ) )
28	15, 6, 7, 4, 26, 1, 27	bnj1253 31085	. . . . . . 7 |- ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) -> { x e. D | ( g ` x ) =/= ( h ` x ) } =/= (/) )
29		nfrab1 3122	. . . . . . . . 9 |- F/_ x { x e. D | ( g ` x ) =/= ( h ` x ) }
30	29	nfcrii 2757	. . . . . . . 8 |- ( z e. { x e. D | ( g ` x ) =/= ( h ` x ) } -> A. x z e. { x e. D | ( g ` x ) =/= ( h ` x ) } )
31	30	bnj1228 31079	. . . . . . 7 |- ( ( R FrSe A /\ { x e. D | ( g ` x ) =/= ( h ` x ) } C_ A /\ { x e. D | ( g ` x ) =/= ( h ` x ) } =/= (/) ) -> E. x e. { x e. D | ( g ` x ) =/= ( h ` x ) } A. y e. { x e. D | ( g ` x ) =/= ( h ` x ) } -. y R x )
32	2, 25, 28, 31	syl3anc 1326	. . . . . 6 |- ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) -> E. x e. { x e. D | ( g ` x ) =/= ( h ` x ) } A. y e. { x e. D | ( g ` x ) =/= ( h ` x ) } -. y R x )
33		ax-5 1839	. . . . . . 7 |- ( R FrSe A -> A. x R FrSe A )
34	15	bnj1309 31090	. . . . . . . . 9 |- ( w e. B -> A. x w e. B )
35	7, 34	bnj1307 31091	. . . . . . . 8 |- ( w e. C -> A. x w e. C )
36	35	hblem 2731	. . . . . . 7 |- ( g e. C -> A. x g e. C )
37	35	hblem 2731	. . . . . . 7 |- ( h e. C -> A. x h e. C )
38		ax-5 1839	. . . . . . 7 |- ( ( g |` D ) =/= ( h |` D ) -> A. x ( g |` D ) =/= ( h |` D ) )
39	33, 36, 37, 38	bnj982 30849	. . . . . 6 |- ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) -> A. x ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
40	32, 27, 39	bnj1521 30921	. . . . 5 |- ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) -> E. x ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) /\ x e. { x e. D | ( g ` x ) =/= ( h ` x ) } /\ A. y e. { x e. D | ( g ` x ) =/= ( h ` x ) } -. y R x ) )
41		simp2 1062	. . . . 5 |- ( ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) /\ x e. { x e. D | ( g ` x ) =/= ( h ` x ) } /\ A. y e. { x e. D | ( g ` x ) =/= ( h ` x ) } -. y R x ) -> x e. { x e. D | ( g ` x ) =/= ( h ` x ) } )
42	15, 6, 7, 4, 26, 1, 27	bnj1279 31086	. . . . . . . . 9 |- ( ( x e. { x e. D | ( g ` x ) =/= ( h ` x ) } /\ A. y e. { x e. D | ( g ` x ) =/= ( h ` x ) } -. y R x ) -> ( pred ( x , A , R ) i^i { x e. D | ( g ` x ) =/= ( h ` x ) } ) = (/) )
43	42	3adant1 1079	. . . . . . . 8 |- ( ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) /\ x e. { x e. D | ( g ` x ) =/= ( h ` x ) } /\ A. y e. { x e. D | ( g ` x ) =/= ( h ` x ) } -. y R x ) -> ( pred ( x , A , R ) i^i { x e. D | ( g ` x ) =/= ( h ` x ) } ) = (/) )
44	15, 6, 7, 4, 26, 1, 27, 43	bnj1280 31088	. . . . . . 7 |- ( ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) /\ x e. { x e. D | ( g ` x ) =/= ( h ` x ) } /\ A. y e. { x e. D | ( g ` x ) =/= ( h ` x ) } -. y R x ) -> ( g |` pred ( x , A , R ) ) = ( h |` pred ( x , A , R ) ) )
45		eqid 2622	. . . . . . 7 |- <. x , ( h |` pred ( x , A , R ) ) >. = <. x , ( h |` pred ( x , A , R ) ) >.
46		eqid 2622	. . . . . . 7 |- { h | E. d e. B ( h Fn d /\ A. x e. d ( h ` x ) = ( G ` <. x , ( h |` pred ( x , A , R ) ) >. ) ) } = { h | E. d e. B ( h Fn d /\ A. x e. d ( h ` x ) = ( G ` <. x , ( h |` pred ( x , A , R ) ) >. ) ) }
47	15, 6, 7, 4, 26, 1, 27, 44, 8, 9, 45, 46	bnj1296 31089	. . . . . 6 |- ( ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) /\ x e. { x e. D | ( g ` x ) =/= ( h ` x ) } /\ A. y e. { x e. D | ( g ` x ) =/= ( h ` x ) } -. y R x ) -> ( g ` x ) = ( h ` x ) )
48	26	bnj1538 30925	. . . . . . 7 |- ( x e. { x e. D | ( g ` x ) =/= ( h ` x ) } -> ( g ` x ) =/= ( h ` x ) )
49	48	necon2bi 2824	. . . . . 6 |- ( ( g ` x ) = ( h ` x ) -> -. x e. { x e. D | ( g ` x ) =/= ( h ` x ) } )
50	47, 49	syl 17	. . . . 5 |- ( ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) /\ x e. { x e. D | ( g ` x ) =/= ( h ` x ) } /\ A. y e. { x e. D | ( g ` x ) =/= ( h ` x ) } -. y R x ) -> -. x e. { x e. D | ( g ` x ) =/= ( h ` x ) } )
51	40, 41, 50	bnj1304 30890	. . . 4 |- -. ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) )
52		df-bnj17 30753	. . . 4 |- ( ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) <-> ( ( R FrSe A /\ g e. C /\ h e. C ) /\ ( g |` D ) =/= ( h |` D ) ) )
53	51, 52	mtbi 312	. . 3 |- -. ( ( R FrSe A /\ g e. C /\ h e. C ) /\ ( g |` D ) =/= ( h |` D ) )
54	53	imnani 439	. 2 |- ( ( R FrSe A /\ g e. C /\ h e. C ) -> -. ( g |` D ) =/= ( h |` D ) )
55		nne 2798	. 2 |- ( -. ( g |` D ) =/= ( h |` D ) <-> ( g |` D ) = ( h |` D ) )
56	54, 55	sylib 208	1 |- ( ( R FrSe A /\ g e. C /\ h e. C ) -> ( g |` D ) = ( h |` D ) )

Syntax hints:   -. wn 3   -> wi 4   /\ 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   (/)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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-reu 2919  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759  df-bnj18 30761  df-bnj19 30763

This theorem is referenced by:  bnj1326  31094  bnj60  31130