Theorem bnj1371 31097

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
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 30910	. . . . . 6 |- ( f e. H -> E. y e. pred ( x , A , R ) ta' )
3		rexex 3002	. . . . . 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 1774	. . . . 5 |- ( E. y ta' <-> E. y ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
7	4, 6	sylib 208	. . . 4 |- ( f e. H -> E. y ( f e. C /\ dom f = ( { y } u. trCl ( y , A , R ) ) ) )
8		exsimpl 1795	. . . 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 2735	. . . . . 6 |- ( f e. C <-> E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
12	11	bnj1238 30877	. . . . 5 |- ( f e. C -> E. d e. B f Fn d )
13		fnfun 5988	. . . . 5 |- ( f Fn d -> Fun f )
14	12, 13	bnj31 30785	. . . 4 |- ( f e. C -> E. d e. B Fun f )
15	14	bnj1265 30883	. . 3 |- ( f e. C -> Fun f )
16	9, 15	bnj593 30815	. 2 |- ( f e. H -> E. y Fun f )
17	16	bnj937 30842	1 |- ( f e. H -> Fun f )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   [.wsbc 3435   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U.cuni 4436   class class class wbr 4653   dom cdm 5114   |` cres 5116   Fun wfun 5882   Fn wfn 5883   ` cfv 5888   predc-bnj14 30754   FrSe w-bnj15 30758   trClc-bnj18 30760

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-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-rex 2918  df-fn 5891

This theorem is referenced by:  bnj1384  31100