Theorem bnj1256 31083

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
bnj1256.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1256.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1256.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1256.4	|- D = ( dom g i^i dom h )
bnj1256.5	|- E = { x e. D | ( g ` x ) =/= ( h ` x ) }
bnj1256.6	|- ( ph <-> ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
bnj1256.7	|- ( ps <-> ( ph /\ x e. E /\ A. y e. E -. y R x ) )

Assertion

Ref	Expression
bnj1256	|- ( ph -> E. d e. B g Fn d )

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

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, h, d)   C( x, y, f, g, h, d)   D( x, y, f, g, h, d)   R( x, y, g, h, d)   E( x, y, f, g, h, d)   G( x, y, h, d)   Y( x, y, f, h, d)

Proof of Theorem bnj1256

Step	Hyp	Ref	Expression
1		bnj1256.6	. 2 |- ( ph <-> ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
2		abid 2610	. . . 4 |- ( 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 ) ) >. ) ) )
3	2	bnj1238 30877	. . 3 |- ( 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 )
4		bnj1256.2	. . . 4 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
5		bnj1256.3	. . . 4 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
6		eqid 2622	. . . 4 |- <. x , ( g |` pred ( x , A , R ) ) >. = <. x , ( g |` pred ( x , A , R ) ) >.
7		eqid 2622	. . . 4 |- { 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 ) ) >. ) ) }
8	4, 5, 6, 7	bnj1234 31081	. . 3 |- C = { g | E. d e. B ( g Fn d /\ A. x e. d ( g ` x ) = ( G ` <. x , ( g |` pred ( x , A , R ) ) >. ) ) }
9	3, 8	eleq2s 2719	. 2 |- ( g e. C -> E. d e. B g Fn d )
10	1, 9	bnj770 30833	1 |- ( ph -> E. d e. B g Fn d )

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:  bnj1253  31085  bnj1286  31087  bnj1280  31088