Theorem bnj1307 31091

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
bnj1307.1	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1307.2	|- ( w e. B -> A. x w e. B )

Assertion

Ref	Expression
bnj1307	|- ( w e. C -> A. x w e. C )

Distinct variable groups:   w, B   w, d, x   x, f

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

Proof of Theorem bnj1307

Step	Hyp	Ref	Expression
1		bnj1307.1	. . 3 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
2		bnj1307.2	. . . . . 6 |- ( w e. B -> A. x w e. B )
3	2	nfcii 2755	. . . . 5 |- F/_ x B
4		nfv 1843	. . . . . 6 |- F/ x f Fn d
5		nfra1 2941	. . . . . 6 |- F/ x A. x e. d ( f ` x ) = ( G ` Y )
6	4, 5	nfan 1828	. . . . 5 |- F/ x ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) )
7	3, 6	nfrex 3007	. . . 4 |- F/ x E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) )
8	7	nfab 2769	. . 3 |- F/_ x { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
9	1, 8	nfcxfr 2762	. 2 |- F/_ x C
10	9	nfcrii 2757	1 |- ( w e. C -> A. x w e. C )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   Fn wfn 5883   ` cfv 5888

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

This theorem is referenced by:  bnj1311  31092  bnj1373  31098  bnj1498  31129  bnj1525  31137  bnj1523  31139