Theorem bnj1371 29910

Description: Technical lemma for bnj60 29943. 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
bnj1371.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1371.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1371.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1371.4	|- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
bnj1371.5	|- D = { x e. A | -. E. f ta }
bnj1371.6	|- ( ps <-> ( R FrSe A /\ D =/= (/) ) )
bnj1371.7	|- ( ch <-> ( ps /\ x e. D /\ A. y e. D -. y R x ) )
bnj1371.8	|- ( ta' <-> [. y / x ]. ta )
bnj1371.9	|- H = { f | E. y e. pred ( x , A , R ) ta' }
bnj1371.10	|- P = U. H
bnj1371.11	|- ( ta' <-> ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )

Assertion

Ref	Expression
bnj1371	|- ( f e. H -> Fun f )

Distinct variable groups:   f, d   y, f

Allowed substitution hints:   ps( x, y, f, d)   ch( x, y, f, d)   ta( x, y, f, d)   A( x, y, f, d)   B( x, y, f, d)   C( x, y, f, d)   D( x, y, f, d)   P( x, y, f, d)   R( x, y, f, d)   G( x, y, f, d)   H( x, y, f, d)   Y( x, y, f, d)   ta'( x, y, f, d)

Proof of Theorem bnj1371

Step	Hyp	Ref	Expression
1		bnj1371.9	. . . . . . 7 |- H = { f | E. y e. pred ( x , A , R ) ta' }
2	1	bnj1436 29723	. . . . . 6 |- ( f e. H -> E. y e. pred ( x , A , R ) ta' )
3		rexex 2843	. . . . . 6 |- ( E. y e. pred ( x , A , R ) ta' -> E. y ta' )
4	2, 3	syl 17	. . . . 5 |- ( f e. H -> E. y ta' )
5		bnj1371.11	. . . . . 6 |- ( ta' <-> ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
6	5	exbii 1726	. . . . 5 |- ( E. y ta' <-> E. y ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
7	4, 6	sylib 201	. . . 4 |- ( f e. H -> E. y ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
8		exsimpl 1737	. . . 4 |- ( E. y ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) -> E. y f e. C )
9	7, 8	syl 17	. . 3 |- ( f e. H -> E. y f e. C )
10		bnj1371.3	. . . . . . 7 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
11	10	abeq2i 2583	. . . . . 6 |- ( f e. C <-> E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
12	11	bnj1238 29690	. . . . 5 |- ( f e. C -> E. d e. B f Fn d )
13		fnfun 5683	. . . . 5 |- ( f Fn d -> Fun f )
14	12, 13	bnj31 29597	. . . 4 |- ( f e. C -> E. d e. B Fun f )
15	14	bnj1265 29696	. . 3 |- ( f e. C -> Fun f )
16	9, 15	bnj593 29627	. 2 |- ( f e. H -> E. y Fun f )
17	16	bnj937 29655	1 |- ( f e. H -> Fun f )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   E.wex 1671   e. wcel 1904   {cab 2457   =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760   [.wsbc 3255   u. cun 3388   C_ wss 3390   (/)c0 3722   {csn 3959   <.cop 3965   U.cuni 4190   class class class wbr 4395   dom cdm 4839   |` cres 4841   Fun wfun 5583   Fn wfn 5584   ` cfv 5589   predc-bnj14 29565   FrSe w-bnj15 29569   trClc-bnj18 29571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-12 1950  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-ral 2761  df-rex 2762  df-fn 5592

This theorem is referenced by:  bnj1384  29913