Theorem bnj1497 31128

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
bnj1497.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1497.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1497.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }

Assertion

Ref	Expression
bnj1497	|- A. g e. C Fun g

Distinct variable groups:   C, g   f, d   f, g

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

Proof of Theorem bnj1497

Step	Hyp	Ref	Expression
1		bnj1497.3	. . . . . 6 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
2	1	bnj1317 30892	. . . . 5 |- ( g e. C -> A. f g e. C )
3	2	nf5i 2024	. . . 4 |- F/ f g e. C
4		nfv 1843	. . . 4 |- F/ f Fun g
5	3, 4	nfim 1825	. . 3 |- F/ f ( g e. C -> Fun g )
6		eleq1 2689	. . . 4 |- ( f = g -> ( f e. C <-> g e. C ) )
7		funeq 5908	. . . 4 |- ( f = g -> ( Fun f <-> Fun g ) )
8	6, 7	imbi12d 334	. . 3 |- ( f = g -> ( ( f e. C -> Fun f ) <-> ( g e. C -> Fun g ) ) )
9	1	bnj1436 30910	. . . . . 6 |- ( f e. C -> E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
10	9	bnj1299 30889	. . . . 5 |- ( f e. C -> E. d e. B f Fn d )
11		fnfun 5988	. . . . 5 |- ( f Fn d -> Fun f )
12	10, 11	bnj31 30785	. . . 4 |- ( f e. C -> E. d e. B Fun f )
13	12	bnj1265 30883	. . 3 |- ( f e. C -> Fun f )
14	5, 8, 13	chvar 2262	. 2 |- ( g e. C -> Fun g )
15	14	rgen 2922	1 |- A. g e. C Fun g

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   C_ wss 3574   <.cop 4183   |` cres 5116   Fun wfun 5882   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-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-in 3581  df-ss 3588  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-fun 5890  df-fn 5891

This theorem is referenced by:  bnj60  31130