Theorem bnj1296 31089

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
bnj1296.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1296.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1296.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1296.4	|- D = ( dom g i^i dom h )
bnj1296.5	|- E = { x e. D | ( g ` x ) =/= ( h ` x ) }
bnj1296.6	|- ( ph <-> ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
bnj1296.7	|- ( ps <-> ( ph /\ x e. E /\ A. y e. E -. y R x ) )
bnj1296.18	|- ( ps -> ( g |` pred ( x , A , R ) ) = ( h |` pred ( x , A , R ) ) )
bnj1296.9	|- Z = <. x , ( g |` pred ( x , A , R ) ) >.
bnj1296.10	|- K = { g | E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` Z ) ) }
bnj1296.11	|- W = <. x , ( h |` pred ( x , A , R ) ) >.
bnj1296.12	|- L = { h | E. d e. B ( h Fn d /\ A. x e. d ( h ` x ) = ( G ` W ) ) }

Assertion

Ref	Expression
bnj1296	|- ( ps -> ( g ` x ) = ( h ` x ) )

Distinct variable groups:   B, f, g   B, h, f   x, D   G, d, f, g   h, G, d   W, d, f   g, Y   h, Y   Z, d, f   x, d, f, g   x, h

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

Proof of Theorem bnj1296

Step	Hyp	Ref	Expression
1		bnj1296.18	. . . . 5 |- ( ps -> ( g |` pred ( x , A , R ) ) = ( h |` pred ( x , A , R ) ) )
2	1	opeq2d 4409	. . . 4 |- ( ps -> <. x , ( g |` pred ( x , A , R ) ) >. = <. x , ( h |` pred ( x , A , R ) ) >. )
3		bnj1296.9	. . . 4 |- Z = <. x , ( g |` pred ( x , A , R ) ) >.
4		bnj1296.11	. . . 4 |- W = <. x , ( h |` pred ( x , A , R ) ) >.
5	2, 3, 4	3eqtr4g 2681	. . 3 |- ( ps -> Z = W )
6	5	fveq2d 6195	. 2 |- ( ps -> ( G ` Z ) = ( G ` W ) )
7		bnj1296.7	. . . 4 |- ( ps <-> ( ph /\ x e. E /\ A. y e. E -. y R x ) )
8		bnj1296.6	. . . . 5 |- ( ph <-> ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
9		bnj1296.10	. . . . . . . . . . 11 |- K = { g | E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` Z ) ) }
10	9	bnj1436 30910	. . . . . . . . . 10 |- ( g e. K -> E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` Z ) ) )
11		fndm 5990	. . . . . . . . . . 11 |- ( g Fn d -> dom g = d )
12	11	anim1i 592	. . . . . . . . . 10 |- ( ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` Z ) ) -> ( dom g = d /\ A. x e. d ( g ` x ) = ( G ` Z ) ) )
13	10, 12	bnj31 30785	. . . . . . . . 9 |- ( g e. K -> E. d e. B ( dom g = d /\ A. x e. d ( g ` x ) = ( G ` Z ) ) )
14		raleq 3138	. . . . . . . . . . 11 |- ( dom g = d -> ( A. x e. dom g ( g ` x ) = ( G ` Z ) <-> A. x e. d ( g ` x ) = ( G ` Z ) ) )
15	14	pm5.32i 669	. . . . . . . . . 10 |- ( ( dom g = d /\ A. x e. dom g ( g ` x ) = ( G ` Z ) ) <-> ( dom g = d /\ A. x e. d ( g ` x ) = ( G ` Z ) ) )
16	15	rexbii 3041	. . . . . . . . 9 |- ( E. d e. B ( dom g = d /\ A. x e. dom g ( g ` x ) = ( G ` Z ) ) <-> E. d e. B ( dom g = d /\ A. x e. d ( g ` x ) = ( G ` Z ) ) )
17	13, 16	sylibr 224	. . . . . . . 8 |- ( g e. K -> E. d e. B ( dom g = d /\ A. x e. dom g ( g ` x ) = ( G ` Z ) ) )
18		simpr 477	. . . . . . . 8 |- ( ( dom g = d /\ A. x e. dom g ( g ` x ) = ( G ` Z ) ) -> A. x e. dom g ( g ` x ) = ( G ` Z ) )
19	17, 18	bnj31 30785	. . . . . . 7 |- ( g e. K -> E. d e. B A. x e. dom g ( g ` x ) = ( G ` Z ) )
20	19	bnj1265 30883	. . . . . 6 |- ( g e. K -> A. x e. dom g ( g ` x ) = ( G ` Z ) )
21		bnj1296.2	. . . . . . 7 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
22		bnj1296.3	. . . . . . 7 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
23	21, 22, 3, 9	bnj1234 31081	. . . . . 6 |- C = K
24	20, 23	eleq2s 2719	. . . . 5 |- ( g e. C -> A. x e. dom g ( g ` x ) = ( G ` Z ) )
25	8, 24	bnj770 30833	. . . 4 |- ( ph -> A. x e. dom g ( g ` x ) = ( G ` Z ) )
26	7, 25	bnj835 30829	. . 3 |- ( ps -> A. x e. dom g ( g ` x ) = ( G ` Z ) )
27		bnj1296.4	. . . . 5 |- D = ( dom g i^i dom h )
28	27	bnj1292 30886	. . . 4 |- D C_ dom g
29		bnj1296.5	. . . . 5 |- E = { x e. D | ( g ` x ) =/= ( h ` x ) }
30	29, 7	bnj1212 30870	. . . 4 |- ( ps -> x e. D )
31	28, 30	bnj1213 30869	. . 3 |- ( ps -> x e. dom g )
32	26, 31	bnj1294 30888	. 2 |- ( ps -> ( g ` x ) = ( G ` Z ) )
33		bnj1296.12	. . . . . . . . . . 11 |- L = { h | E. d e. B ( h Fn d /\ A. x e. d ( h ` x ) = ( G ` W ) ) }
34	33	bnj1436 30910	. . . . . . . . . 10 |- ( h e. L -> E. d e. B ( h Fn d /\ A. x e. d ( h ` x ) = ( G ` W ) ) )
35		fndm 5990	. . . . . . . . . . 11 |- ( h Fn d -> dom h = d )
36	35	anim1i 592	. . . . . . . . . 10 |- ( ( h Fn d /\ A. x e. d ( h ` x ) = ( G ` W ) ) -> ( dom h = d /\ A. x e. d ( h ` x ) = ( G ` W ) ) )
37	34, 36	bnj31 30785	. . . . . . . . 9 |- ( h e. L -> E. d e. B ( dom h = d /\ A. x e. d ( h ` x ) = ( G ` W ) ) )
38		raleq 3138	. . . . . . . . . . 11 |- ( dom h = d -> ( A. x e. dom h ( h ` x ) = ( G ` W ) <-> A. x e. d ( h ` x ) = ( G ` W ) ) )
39	38	pm5.32i 669	. . . . . . . . . 10 |- ( ( dom h = d /\ A. x e. dom h ( h ` x ) = ( G ` W ) ) <-> ( dom h = d /\ A. x e. d ( h ` x ) = ( G ` W ) ) )
40	39	rexbii 3041	. . . . . . . . 9 |- ( E. d e. B ( dom h = d /\ A. x e. dom h ( h ` x ) = ( G ` W ) ) <-> E. d e. B ( dom h = d /\ A. x e. d ( h ` x ) = ( G ` W ) ) )
41	37, 40	sylibr 224	. . . . . . . 8 |- ( h e. L -> E. d e. B ( dom h = d /\ A. x e. dom h ( h ` x ) = ( G ` W ) ) )
42		simpr 477	. . . . . . . 8 |- ( ( dom h = d /\ A. x e. dom h ( h ` x ) = ( G ` W ) ) -> A. x e. dom h ( h ` x ) = ( G ` W ) )
43	41, 42	bnj31 30785	. . . . . . 7 |- ( h e. L -> E. d e. B A. x e. dom h ( h ` x ) = ( G ` W ) )
44	43	bnj1265 30883	. . . . . 6 |- ( h e. L -> A. x e. dom h ( h ` x ) = ( G ` W ) )
45	21, 22, 4, 33	bnj1234 31081	. . . . . 6 |- C = L
46	44, 45	eleq2s 2719	. . . . 5 |- ( h e. C -> A. x e. dom h ( h ` x ) = ( G ` W ) )
47	8, 46	bnj771 30834	. . . 4 |- ( ph -> A. x e. dom h ( h ` x ) = ( G ` W ) )
48	7, 47	bnj835 30829	. . 3 |- ( ps -> A. x e. dom h ( h ` x ) = ( G ` W ) )
49	27	bnj1293 30887	. . . 4 |- D C_ dom h
50	49, 30	bnj1213 30869	. . 3 |- ( ps -> x e. dom h )
51	48, 50	bnj1294 30888	. 2 |- ( ps -> ( h ` x ) = ( G ` W ) )
52	6, 32, 51	3eqtr4d 2666	1 |- ( ps -> ( g ` x ) = ( h ` x ) )

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:  bnj1311  31092