Theorem bnj1520 31134

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
bnj1520.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1520.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1520.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1520.4	|- F = U. C

Assertion

Ref	Expression
bnj1520	|- ( ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) -> A. f ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) )

Distinct variable groups:   A, f   f, G   R, f   x, f

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

Proof of Theorem bnj1520

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1520.4	. . . . 5 |- F = U. C
2		bnj1520.3	. . . . . . . 8 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
3	2	bnj1317 30892	. . . . . . 7 |- ( w e. C -> A. f w e. C )
4	3	nfcii 2755	. . . . . 6 |- F/_ f C
5	4	nfuni 4442	. . . . 5 |- F/_ f U. C
6	1, 5	nfcxfr 2762	. . . 4 |- F/_ f F
7		nfcv 2764	. . . 4 |- F/_ f x
8	6, 7	nffv 6198	. . 3 |- F/_ f ( F ` x )
9		nfcv 2764	. . . 4 |- F/_ f G
10		nfcv 2764	. . . . . 6 |- F/_ f pred ( x , A , R )
11	6, 10	nfres 5398	. . . . 5 |- F/_ f ( F |` pred ( x , A , R ) )
12	7, 11	nfop 4418	. . . 4 |- F/_ f <. x , ( F |` pred ( x , A , R ) ) >.
13	9, 12	nffv 6198	. . 3 |- F/_ f ( G ` <. x , ( F |` pred ( x , A , R ) ) >. )
14	8, 13	nfeq 2776	. 2 |- F/ f ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. )
15	14	nf5ri 2065	1 |- ( ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) -> A. f ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   {cab 2608   A.wral 2912   E.wrex 2913   C_ wss 3574   <.cop 4183   U.cuni 4436   |` 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-13 2246  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-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-res 5126  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj1501  31135