Theorem bnj153 30950

Description: Technical lemma for bnj852 30991. 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
bnj153.1	|- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
bnj153.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj153.3	|- D = ( om \ { (/) } )
bnj153.4	|- ( th <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
bnj153.5	|- ( ta <-> A. m e. D ( m _E n -> [. m / n ]. th ) )

Assertion

Ref	Expression
bnj153	|- ( n = 1o -> ( ( n e. D /\ ta ) -> th ) )

Distinct variable groups:   A, f, i, x, y, n   R, f, i, x, y, n   m, n

Allowed substitution hints:   ph( x, y, f, i, m, n)   ps( x, y, f, i, m, n)   th( x, y, f, i, m, n)   ta( x, y, f, i, m, n)   A( m)   D( x, y, f, i, m, n)   R( m)

Proof of Theorem bnj153

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj153.1	. 2 |- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
2		bnj153.2	. 2 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
3		bnj153.3	. 2 |- D = ( om \ { (/) } )
4		bnj153.4	. 2 |- ( th <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
5		bnj153.5	. 2 |- ( ta <-> A. m e. D ( m _E n -> [. m / n ]. th ) )
6		biid 251	. 2 |- ( ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) <-> ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) )
7		biid 251	. . . 4 |- ( [. 1o / n ]. ph <-> [. 1o / n ]. ph )
8	1, 7	bnj118 30939	. . 3 |- ( [. 1o / n ]. ph <-> ( f ` (/) ) = pred ( x , A , R ) )
9	8	bicomi 214	. 2 |- ( ( f ` (/) ) = pred ( x , A , R ) <-> [. 1o / n ]. ph )
10		bnj105 30790	. . . 4 |- 1o e. _V
11	2, 10	bnj92 30932	. . 3 |- ( [. 1o / n ]. ps <-> A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
12	11	bicomi 214	. 2 |- ( A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) <-> [. 1o / n ]. ps )
13		biid 251	. 2 |- ( [. 1o / n ]. th <-> [. 1o / n ]. th )
14		biid 251	. 2 |- ( ( ( R FrSe A /\ x e. A ) -> E. f ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) ) <-> ( ( R FrSe A /\ x e. A ) -> E. f ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) ) )
15		biid 251	. 2 |- ( ( ( R FrSe A /\ x e. A ) -> E* f ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) ) <-> ( ( R FrSe A /\ x e. A ) -> E* f ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) ) )
16		biid 251	. . . . 5 |- ( [. 1o / n ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) <-> [. 1o / n ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) )
17		biid 251	. . . . 5 |- ( [. 1o / n ]. ps <-> [. 1o / n ]. ps )
18	6, 16, 7, 17	bnj121 30940	. . . 4 |- ( [. 1o / n ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) <-> ( ( R FrSe A /\ x e. A ) -> ( f Fn 1o /\ [. 1o / n ]. ph /\ [. 1o / n ]. ps ) ) )
19	8	anbi2i 730	. . . . . . 7 |- ( ( f Fn 1o /\ [. 1o / n ]. ph ) <-> ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) ) )
20	19, 11	anbi12i 733	. . . . . 6 |- ( ( ( f Fn 1o /\ [. 1o / n ]. ph ) /\ [. 1o / n ]. ps ) <-> ( ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) ) /\ A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
21		df-3an 1039	. . . . . 6 |- ( ( f Fn 1o /\ [. 1o / n ]. ph /\ [. 1o / n ]. ps ) <-> ( ( f Fn 1o /\ [. 1o / n ]. ph ) /\ [. 1o / n ]. ps ) )
22		df-3an 1039	. . . . . 6 |- ( ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) <-> ( ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) ) /\ A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
23	20, 21, 22	3bitr4i 292	. . . . 5 |- ( ( f Fn 1o /\ [. 1o / n ]. ph /\ [. 1o / n ]. ps ) <-> ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
24	23	imbi2i 326	. . . 4 |- ( ( ( R FrSe A /\ x e. A ) -> ( f Fn 1o /\ [. 1o / n ]. ph /\ [. 1o / n ]. ps ) ) <-> ( ( R FrSe A /\ x e. A ) -> ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) ) )
25	18, 24	bitri 264	. . 3 |- ( [. 1o / n ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) <-> ( ( R FrSe A /\ x e. A ) -> ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) ) )
26	25	bicomi 214	. 2 |- ( ( ( R FrSe A /\ x e. A ) -> ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) ) <-> [. 1o / n ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) )
27		eqid 2622	. 2 |- { <. (/) , pred ( x , A , R ) >. } = { <. (/) , pred ( x , A , R ) >. }
28		biid 251	. 2 |- ( [. { <. (/) , pred ( x , A , R ) >. } / f ]. ( f ` (/) ) = pred ( x , A , R ) <-> [. { <. (/) , pred ( x , A , R ) >. } / f ]. ( f ` (/) ) = pred ( x , A , R ) )
29		biid 251	. 2 |- ( [. { <. (/) , pred ( x , A , R ) >. } / f ]. A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) <-> [. { <. (/) , pred ( x , A , R ) >. } / f ]. A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
30	26	sbcbii 3491	. . 3 |- ( [. { <. (/) , pred ( x , A , R ) >. } / f ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) ) <-> [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) )
31		biid 251	. . . . 5 |- ( [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ph <-> [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ph )
32		biid 251	. . . . 5 |- ( [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ps <-> [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ps )
33		biid 251	. . . . 5 |- ( [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) <-> [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) )
34	27, 31, 32, 33, 18	bnj124 30941	. . . 4 |- ( [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) <-> ( ( R FrSe A /\ x e. A ) -> ( { <. (/) , pred ( x , A , R ) >. } Fn 1o /\ [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ph /\ [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ps ) ) )
35	1, 7, 31, 27	bnj125 30942	. . . . . . . 8 |- ( [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ph <-> ( { <. (/) , pred ( x , A , R ) >. } ` (/) ) = pred ( x , A , R ) )
36	35	anbi2i 730	. . . . . . 7 |- ( ( { <. (/) , pred ( x , A , R ) >. } Fn 1o /\ [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ph ) <-> ( { <. (/) , pred ( x , A , R ) >. } Fn 1o /\ ( { <. (/) , pred ( x , A , R ) >. } ` (/) ) = pred ( x , A , R ) ) )
37	2, 17, 32, 27	bnj126 30943	. . . . . . 7 |- ( [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ps <-> A. i e. om ( suc i e. 1o -> ( { <. (/) , pred ( x , A , R ) >. } ` suc i ) = U_ y e. ( { <. (/) , pred ( x , A , R ) >. } ` i ) pred ( y , A , R ) ) )
38	36, 37	anbi12i 733	. . . . . 6 |- ( ( ( { <. (/) , pred ( x , A , R ) >. } Fn 1o /\ [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ph ) /\ [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ps ) <-> ( ( { <. (/) , pred ( x , A , R ) >. } Fn 1o /\ ( { <. (/) , pred ( x , A , R ) >. } ` (/) ) = pred ( x , A , R ) ) /\ A. i e. om ( suc i e. 1o -> ( { <. (/) , pred ( x , A , R ) >. } ` suc i ) = U_ y e. ( { <. (/) , pred ( x , A , R ) >. } ` i ) pred ( y , A , R ) ) ) )
39		df-3an 1039	. . . . . 6 |- ( ( { <. (/) , pred ( x , A , R ) >. } Fn 1o /\ [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ph /\ [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ps ) <-> ( ( { <. (/) , pred ( x , A , R ) >. } Fn 1o /\ [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ph ) /\ [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ps ) )
40		df-3an 1039	. . . . . 6 |- ( ( { <. (/) , pred ( x , A , R ) >. } Fn 1o /\ ( { <. (/) , pred ( x , A , R ) >. } ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( { <. (/) , pred ( x , A , R ) >. } ` suc i ) = U_ y e. ( { <. (/) , pred ( x , A , R ) >. } ` i ) pred ( y , A , R ) ) ) <-> ( ( { <. (/) , pred ( x , A , R ) >. } Fn 1o /\ ( { <. (/) , pred ( x , A , R ) >. } ` (/) ) = pred ( x , A , R ) ) /\ A. i e. om ( suc i e. 1o -> ( { <. (/) , pred ( x , A , R ) >. } ` suc i ) = U_ y e. ( { <. (/) , pred ( x , A , R ) >. } ` i ) pred ( y , A , R ) ) ) )
41	38, 39, 40	3bitr4i 292	. . . . 5 |- ( ( { <. (/) , pred ( x , A , R ) >. } Fn 1o /\ [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ph /\ [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ps ) <-> ( { <. (/) , pred ( x , A , R ) >. } Fn 1o /\ ( { <. (/) , pred ( x , A , R ) >. } ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( { <. (/) , pred ( x , A , R ) >. } ` suc i ) = U_ y e. ( { <. (/) , pred ( x , A , R ) >. } ` i ) pred ( y , A , R ) ) ) )
42	41	imbi2i 326	. . . 4 |- ( ( ( R FrSe A /\ x e. A ) -> ( { <. (/) , pred ( x , A , R ) >. } Fn 1o /\ [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ph /\ [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ps ) ) <-> ( ( R FrSe A /\ x e. A ) -> ( { <. (/) , pred ( x , A , R ) >. } Fn 1o /\ ( { <. (/) , pred ( x , A , R ) >. } ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( { <. (/) , pred ( x , A , R ) >. } ` suc i ) = U_ y e. ( { <. (/) , pred ( x , A , R ) >. } ` i ) pred ( y , A , R ) ) ) ) )
43	34, 42	bitri 264	. . 3 |- ( [. { <. (/) , pred ( x , A , R ) >. } / f ]. [. 1o / n ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) <-> ( ( R FrSe A /\ x e. A ) -> ( { <. (/) , pred ( x , A , R ) >. } Fn 1o /\ ( { <. (/) , pred ( x , A , R ) >. } ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( { <. (/) , pred ( x , A , R ) >. } ` suc i ) = U_ y e. ( { <. (/) , pred ( x , A , R ) >. } ` i ) pred ( y , A , R ) ) ) ) )
44	30, 43	bitr2i 265	. 2 |- ( ( ( R FrSe A /\ x e. A ) -> ( { <. (/) , pred ( x , A , R ) >. } Fn 1o /\ ( { <. (/) , pred ( x , A , R ) >. } ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( { <. (/) , pred ( x , A , R ) >. } ` suc i ) = U_ y e. ( { <. (/) , pred ( x , A , R ) >. } ` i ) pred ( y , A , R ) ) ) ) <-> [. { <. (/) , pred ( x , A , R ) >. } / f ]. ( ( R FrSe A /\ x e. A ) -> ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) ) )
45		biid 251	. 2 |- ( ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) <-> ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
46		biid 251	. . . . 5 |- ( ( f Fn 1o /\ [. 1o / n ]. ph /\ [. 1o / n ]. ps ) <-> ( f Fn 1o /\ [. 1o / n ]. ph /\ [. 1o / n ]. ps ) )
47		biid 251	. . . . 5 |- ( [. g / f ]. ( f Fn 1o /\ [. 1o / n ]. ph /\ [. 1o / n ]. ps ) <-> [. g / f ]. ( f Fn 1o /\ [. 1o / n ]. ph /\ [. 1o / n ]. ps ) )
48		biid 251	. . . . 5 |- ( [. g / f ]. [. 1o / n ]. ph <-> [. g / f ]. [. 1o / n ]. ph )
49		biid 251	. . . . 5 |- ( [. g / f ]. [. 1o / n ]. ps <-> [. g / f ]. [. 1o / n ]. ps )
50	46, 47, 48, 49	bnj156 30796	. . . 4 |- ( [. g / f ]. ( f Fn 1o /\ [. 1o / n ]. ph /\ [. 1o / n ]. ps ) <-> ( g Fn 1o /\ [. g / f ]. [. 1o / n ]. ph /\ [. g / f ]. [. 1o / n ]. ps ) )
51	48, 8	bnj154 30948	. . . . . . 7 |- ( [. g / f ]. [. 1o / n ]. ph <-> ( g ` (/) ) = pred ( x , A , R ) )
52	51	anbi2i 730	. . . . . 6 |- ( ( g Fn 1o /\ [. g / f ]. [. 1o / n ]. ph ) <-> ( g Fn 1o /\ ( g ` (/) ) = pred ( x , A , R ) ) )
53	17, 11	bitri 264	. . . . . . 7 |- ( [. 1o / n ]. ps <-> A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
54	49, 53	bnj155 30949	. . . . . 6 |- ( [. g / f ]. [. 1o / n ]. ps <-> A. i e. om ( suc i e. 1o -> ( g ` suc i ) = U_ y e. ( g ` i ) pred ( y , A , R ) ) )
55	52, 54	anbi12i 733	. . . . 5 |- ( ( ( g Fn 1o /\ [. g / f ]. [. 1o / n ]. ph ) /\ [. g / f ]. [. 1o / n ]. ps ) <-> ( ( g Fn 1o /\ ( g ` (/) ) = pred ( x , A , R ) ) /\ A. i e. om ( suc i e. 1o -> ( g ` suc i ) = U_ y e. ( g ` i ) pred ( y , A , R ) ) ) )
56		df-3an 1039	. . . . 5 |- ( ( g Fn 1o /\ [. g / f ]. [. 1o / n ]. ph /\ [. g / f ]. [. 1o / n ]. ps ) <-> ( ( g Fn 1o /\ [. g / f ]. [. 1o / n ]. ph ) /\ [. g / f ]. [. 1o / n ]. ps ) )
57		df-3an 1039	. . . . 5 |- ( ( g Fn 1o /\ ( g ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( g ` suc i ) = U_ y e. ( g ` i ) pred ( y , A , R ) ) ) <-> ( ( g Fn 1o /\ ( g ` (/) ) = pred ( x , A , R ) ) /\ A. i e. om ( suc i e. 1o -> ( g ` suc i ) = U_ y e. ( g ` i ) pred ( y , A , R ) ) ) )
58	55, 56, 57	3bitr4i 292	. . . 4 |- ( ( g Fn 1o /\ [. g / f ]. [. 1o / n ]. ph /\ [. g / f ]. [. 1o / n ]. ps ) <-> ( g Fn 1o /\ ( g ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( g ` suc i ) = U_ y e. ( g ` i ) pred ( y , A , R ) ) ) )
59	50, 58	bitri 264	. . 3 |- ( [. g / f ]. ( f Fn 1o /\ [. 1o / n ]. ph /\ [. 1o / n ]. ps ) <-> ( g Fn 1o /\ ( g ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( g ` suc i ) = U_ y e. ( g ` i ) pred ( y , A , R ) ) ) )
60	23	sbcbii 3491	. . 3 |- ( [. g / f ]. ( f Fn 1o /\ [. 1o / n ]. ph /\ [. 1o / n ]. ps ) <-> [. g / f ]. ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
61	59, 60	bitr3i 266	. 2 |- ( ( g Fn 1o /\ ( g ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( g ` suc i ) = U_ y e. ( g ` i ) pred ( y , A , R ) ) ) <-> [. g / f ]. ( f Fn 1o /\ ( f ` (/) ) = pred ( x , A , R ) /\ A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
62		biid 251	. 2 |- ( [. g / f ]. ( f ` (/) ) = pred ( x , A , R ) <-> [. g / f ]. ( f ` (/) ) = pred ( x , A , R ) )
63		biid 251	. 2 |- ( [. g / f ]. A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) <-> [. g / f ]. A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
64	1, 2, 3, 4, 5, 6, 9, 12, 13, 14, 15, 26, 27, 28, 29, 44, 45, 61, 62, 63	bnj151 30947	1 |- ( n = 1o -> ( ( n e. D /\ ta ) -> th ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   E*wmo 2471   A.wral 2912   [.wsbc 3435   \ cdif 3571   (/)c0 3915   {csn 4177   <.cop 4183   U_ciun 4520   class class class wbr 4653   _E cep 5028   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   1oc1o 7553   predc-bnj14 30754   FrSe w-bnj15 30758

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  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1o 7560  df-bnj13 30757  df-bnj15 30759

This theorem is referenced by:  bnj852  30991