Theorem bnj1259 32310

Description: Technical lemma for bnj60 32356. 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 2438	. . . 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 32103	. . 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 2451	. . . 4 |- <. x , ( h |` pred ( x , A , R ) ) >. = <. x , ( h |` pred ( x , A , R ) ) >.
7		eqid 2451	. . . 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 32307	. . 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 2559	. 2 |- ( h e. C -> E. d e. B h Fn d )
10	1, 9	bnj771 32060	1 |- ( ph -> E. d e. B h Fn d )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   {cab 2436   =/= wne 2644   A.wral 2795   E.wrex 2796   {crab 2799   i^i cin 3428   C_ wss 3429   <.cop 3984   class class class wbr 4393   dom cdm 4941   |` cres 4943   Fn wfn 5514   ` cfv 5519   /\ w-bnj17 31977   predc-bnj14 31979   FrSe w-bnj15 31983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-res 4953  df-iota 5482  df-fun 5521  df-fn 5522  df-fv 5527  df-bnj17 31978

This theorem is referenced by:  bnj1253  32311  bnj1286  32313  bnj1280  32314