Theorem bnj958 31010

Description: Technical lemma for bnj69 31078. 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
bnj958.1	|- C = U_ y e. ( f ` m ) pred ( y , A , R )
bnj958.2	|- G = ( f u. { <. n , C >. } )

Assertion

Ref	Expression
bnj958	|- ( ( G ` i ) = ( f ` i ) -> A. y ( G ` i ) = ( f ` i ) )

Distinct variable groups:   y, f   y, i   y, n

Allowed substitution hints:   A( y, f, i, m, n)   C( y, f, i, m, n)   R( y, f, i, m, n)   G( y, f, i, m, n)

Proof of Theorem bnj958

Step	Hyp	Ref	Expression
1		bnj958.2	. . . . 5 |- G = ( f u. { <. n , C >. } )
2		nfcv 2764	. . . . . 6 |- F/_ y f
3		nfcv 2764	. . . . . . . 8 |- F/_ y n
4		bnj958.1	. . . . . . . . 9 |- C = U_ y e. ( f ` m ) pred ( y , A , R )
5		nfiu1 4550	. . . . . . . . 9 |- F/_ y U_ y e. ( f ` m ) pred ( y , A , R )
6	4, 5	nfcxfr 2762	. . . . . . . 8 |- F/_ y C
7	3, 6	nfop 4418	. . . . . . 7 |- F/_ y <. n , C >.
8	7	nfsn 4242	. . . . . 6 |- F/_ y { <. n , C >. }
9	2, 8	nfun 3769	. . . . 5 |- F/_ y ( f u. { <. n , C >. } )
10	1, 9	nfcxfr 2762	. . . 4 |- F/_ y G
11		nfcv 2764	. . . 4 |- F/_ y i
12	10, 11	nffv 6198	. . 3 |- F/_ y ( G ` i )
13	12	nfeq1 2778	. 2 |- F/ y ( G ` i ) = ( f ` i )
14	13	nf5ri 2065	1 |- ( ( G ` i ) = ( f ` i ) -> A. y ( G ` i ) = ( f ` i ) )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   u. cun 3572   {csn 4177   <.cop 4183   U_ciun 4520   ` 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-iun 4522  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj966  31014  bnj967  31015