Theorem bnj1529 31138

Description: Technical lemma for bnj1522 31140. 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
bnj1529.1	|- ( ch -> A. x e. A ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) )
bnj1529.2	|- ( w e. F -> A. x w e. F )

Assertion

Ref	Expression
bnj1529	|- ( ch -> A. y e. A ( F ` y ) = ( G ` <. y , ( F |` pred ( y , A , R ) ) >. ) )

Distinct variable groups:   w, A, x, y   w, F, y   w, G, x, y   w, R, x, y

Allowed substitution hints:   ch( x, y, w)   F( x)

Proof of Theorem bnj1529

Step	Hyp	Ref	Expression
1		bnj1529.1	. 2 |- ( ch -> A. x e. A ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) )
2		nfv 1843	. . 3 |- F/ y ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. )
3		bnj1529.2	. . . . . 6 |- ( w e. F -> A. x w e. F )
4	3	nfcii 2755	. . . . 5 |- F/_ x F
5		nfcv 2764	. . . . 5 |- F/_ x y
6	4, 5	nffv 6198	. . . 4 |- F/_ x ( F ` y )
7		nfcv 2764	. . . . 5 |- F/_ x G
8		nfcv 2764	. . . . . . 7 |- F/_ x pred ( y , A , R )
9	4, 8	nfres 5398	. . . . . 6 |- F/_ x ( F |` pred ( y , A , R ) )
10	5, 9	nfop 4418	. . . . 5 |- F/_ x <. y , ( F |` pred ( y , A , R ) ) >.
11	7, 10	nffv 6198	. . . 4 |- F/_ x ( G ` <. y , ( F |` pred ( y , A , R ) ) >. )
12	6, 11	nfeq 2776	. . 3 |- F/ x ( F ` y ) = ( G ` <. y , ( F |` pred ( y , A , R ) ) >. )
13		fveq2 6191	. . . 4 |- ( x = y -> ( F ` x ) = ( F ` y ) )
14		id 22	. . . . . 6 |- ( x = y -> x = y )
15		bnj602 30985	. . . . . . 7 |- ( x = y -> pred ( x , A , R ) = pred ( y , A , R ) )
16	15	reseq2d 5396	. . . . . 6 |- ( x = y -> ( F |` pred ( x , A , R ) ) = ( F |` pred ( y , A , R ) ) )
17	14, 16	opeq12d 4410	. . . . 5 |- ( x = y -> <. x , ( F |` pred ( x , A , R ) ) >. = <. y , ( F |` pred ( y , A , R ) ) >. )
18	17	fveq2d 6195	. . . 4 |- ( x = y -> ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) = ( G ` <. y , ( F |` pred ( y , A , R ) ) >. ) )
19	13, 18	eqeq12d 2637	. . 3 |- ( x = y -> ( ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) <-> ( F ` y ) = ( G ` <. y , ( F |` pred ( y , A , R ) ) >. ) ) )
20	2, 12, 19	cbvral 3167	. 2 |- ( A. x e. A ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) <-> A. y e. A ( F ` y ) = ( G ` <. y , ( F |` pred ( y , A , R ) ) >. ) )
21	1, 20	sylib 208	1 |- ( ch -> A. y e. A ( F ` y ) = ( G ` <. y , ( F |` pred ( y , A , R ) ) >. ) )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   |` cres 5116   ` 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  df-bnj14 30755

This theorem is referenced by:  bnj1523  31139