Theorem bnj66 30930

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
bnj66.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj66.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj66.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }

Assertion

Ref	Expression
bnj66	|- ( g e. C -> Rel g )

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

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

Proof of Theorem bnj66

Step	Hyp	Ref	Expression
1		bnj66.3	. . . 4 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
2		fneq1 5979	. . . . . . 7 |- ( g = f -> ( g Fn d <-> f Fn d ) )
3		fveq1 6190	. . . . . . . . 9 |- ( g = f -> ( g ` x ) = ( f ` x ) )
4		reseq1 5390	. . . . . . . . . . . 12 |- ( g = f -> ( g |` pred ( x , A , R ) ) = ( f |` pred ( x , A , R ) ) )
5	4	opeq2d 4409	. . . . . . . . . . 11 |- ( g = f -> <. x , ( g |` pred ( x , A , R ) ) >. = <. x , ( f |` pred ( x , A , R ) ) >. )
6		bnj66.2	. . . . . . . . . . 11 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
7	5, 6	syl6eqr 2674	. . . . . . . . . 10 |- ( g = f -> <. x , ( g |` pred ( x , A , R ) ) >. = Y )
8	7	fveq2d 6195	. . . . . . . . 9 |- ( g = f -> ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) = ( G ` Y ) )
9	3, 8	eqeq12d 2637	. . . . . . . 8 |- ( g = f -> ( ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) <-> ( f ` x ) = ( G ` Y ) ) )
10	9	ralbidv 2986	. . . . . . 7 |- ( g = f -> ( A. x e. d ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) <-> A. x e. d ( f ` x ) = ( G ` Y ) ) )
11	2, 10	anbi12d 747	. . . . . 6 |- ( g = f -> ( ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) ) <-> ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) ) )
12	11	rexbidv 3052	. . . . 5 |- ( g = f -> ( E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) ) <-> E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) ) )
13	12	cbvabv 2747	. . . 4 |- { g | E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) ) } = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
14	1, 13	eqtr4i 2647	. . 3 |- C = { g | E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) ) }
15	14	bnj1436 30910	. 2 |- ( g e. C -> E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) ) )
16		bnj1239 30876	. 2 |- ( 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 )
17		fnrel 5989	. . 3 |- ( g Fn d -> Rel g )
18	17	rexlimivw 3029	. 2 |- ( E. d e. B g Fn d -> Rel g )
19	15, 16, 18	3syl 18	1 |- ( g e. C -> Rel g )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   C_ wss 3574   <.cop 4183   |` cres 5116   Rel wrel 5119   Fn wfn 5883   ` cfv 5888   predc-bnj14 30754

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

This theorem is referenced by:  bnj1321  31095