Theorem bnj1259 34215

Description: Technical lemma for bnj60 34261. 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
bnj1259.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1259.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1259.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1259.4	|- D = ( dom g i^i dom h )
bnj1259.5	|- E = { x e. D | ( g ` x ) =/= ( h ` x ) }
bnj1259.6	|- ( ph <-> ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
bnj1259.7	|- ( ps <-> ( ph /\ x e. E /\ A. y e. E -. y R x ) )

Assertion

Ref	Expression
bnj1259	|- ( ph -> E. d e. B h Fn d )

Distinct variable groups:   A, f   B, f, h   f, G, h   R, f   h, Y   f, d, h   x, f, 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, g, 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, g, d)   Y( x, y, f, g, d)

Proof of Theorem bnj1259

Step	Hyp	Ref	Expression
1		bnj1259.6	. 2 |- ( ph <-> ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
2		abid 2444	. . . 4 |- ( h e. { h | E. d e. B ( h Fn d /\ A. x e. d ( h ` x ) = ( G ` <. x , ( h |` pred ( x , A , R ) ) >. ) ) } <-> E. d e. B ( h Fn d /\ A. x e. d ( h ` x ) = ( G ` <. x , ( h |` pred ( x , A , R ) ) >. ) ) )
3	2	bnj1238 34008	. . 3 |- ( h e. { h | E. d e. B ( h Fn d /\ A. x e. d ( h ` x ) = ( G ` <. x , ( h |` pred ( x , A , R ) ) >. ) ) } -> E. d e. B h Fn d )
4		bnj1259.2	. . . 4 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
5		bnj1259.3	. . . 4 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
6		eqid 2457	. . . 4 |- <. x , ( h |` pred ( x , A , R ) ) >. = <. x , ( h |` pred ( x , A , R ) ) >.
7		eqid 2457	. . . 4 |- { 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 ) ) >. ) ) }
8	4, 5, 6, 7	bnj1234 34212	. . 3 |- C = { h | E. d e. B ( h Fn d /\ A. x e. d ( h ` x ) = ( G ` <. x , ( h |` pred ( x , A , R ) ) >. ) ) }
9	3, 8	eleq2s 2565	. 2 |- ( h e. C -> E. d e. B h Fn d )
10	1, 9	bnj771 33965	1 |- ( ph -> E. d e. B h Fn d )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   {cab 2442   =/= wne 2652   A.wral 2807   E.wrex 2808   {crab 2811   i^i cin 3470   C_ wss 3471   <.cop 4038   class class class wbr 4456   dom cdm 5008   |` cres 5010   Fn wfn 5589   ` cfv 5594   /\ w-bnj17 33881   predc-bnj14 33883   FrSe w-bnj15 33887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-res 5020  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-bnj17 33882

This theorem is referenced by:  bnj1253  34216  bnj1286  34218  bnj1280  34219