Theorem bnj1514 31131

Description: Technical lemma for bnj1500 31136. 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
bnj1514.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1514.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1514.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }

Assertion

Ref	Expression
bnj1514	|- ( f e. C -> A. x e. dom f ( f ` x ) = ( G ` Y ) )

Distinct variable groups:   x, A   G, d   Y, d   f, d, x

Allowed substitution hints:   A( f, d)   B( x, f, d)   C( x, f, d)   R( x, f, d)   G( x, f)   Y( x, f)

Proof of Theorem bnj1514

Step	Hyp	Ref	Expression
1		bnj1514.3	. . . . 5 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
2	1	bnj1436 30910	. . . 4 |- ( f e. C -> E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
3		df-rex 2918	. . . . 5 |- ( E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) <-> E. d ( d e. B /\ ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) ) )
4		3anass 1042	. . . . 5 |- ( ( d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) <-> ( d e. B /\ ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) ) )
5	3, 4	bnj133 30793	. . . 4 |- ( E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) <-> E. d ( d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
6	2, 5	sylib 208	. . 3 |- ( f e. C -> E. d ( d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
7		simp3 1063	. . . 4 |- ( ( d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) -> A. x e. d ( f ` x ) = ( G ` Y ) )
8		fndm 5990	. . . . . 6 |- ( f Fn d -> dom f = d )
9	8	3ad2ant2 1083	. . . . 5 |- ( ( d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) -> dom f = d )
10	9	raleqdv 3144	. . . 4 |- ( ( d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) -> ( A. x e. dom f ( f ` x ) = ( G ` Y ) <-> A. x e. d ( f ` x ) = ( G ` Y ) ) )
11	7, 10	mpbird 247	. . 3 |- ( ( d e. B /\ f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) -> A. x e. dom f ( f ` x ) = ( G ` Y ) )
12	6, 11	bnj593 30815	. 2 |- ( f e. C -> E. d A. x e. dom f ( f ` x ) = ( G ` Y ) )
13	12	bnj937 30842	1 |- ( f e. C -> A. x e. dom f ( f ` x ) = ( G ` Y ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   C_ wss 3574   <.cop 4183   dom cdm 5114   |` cres 5116   Fn wfn 5883   ` cfv 5888   predc-bnj14 30754

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-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-fn 5891

This theorem is referenced by:  bnj1501  31135