Theorem bnj580 34318

Description: Technical lemma for bnj579 34319. 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
bnj580.1	|- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
bnj580.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj580.3	|- ( ch <-> ( f Fn n /\ ph /\ ps ) )
bnj580.4	|- ( ph' <-> [. g / f ]. ph )
bnj580.5	|- ( ps' <-> [. g / f ]. ps )
bnj580.6	|- ( ch' <-> [. g / f ]. ch )
bnj580.7	|- D = ( om \ { (/) } )
bnj580.8	|- ( th <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
bnj580.9	|- ( ta <-> A. k e. n ( k _E j -> [. k / j ]. th ) )

Assertion

Ref	Expression
bnj580	|- ( n e. D -> E* f ch )

Distinct variable groups:   A, f, i, k   D, f, g, j, k   R, f, i, k   ch, g, j, k   j, ch', k   f, n   g, i, n, k   x, f   y, f, g, i, k   j, n   th, k

Allowed substitution hints:   ph( x, y, f, g, i, j, k, n)   ps( x, y, f, g, i, j, k, n)   ch( x, y, f, i, n)   th( x, y, f, g, i, j, n)   ta( x, y, f, g, i, j, k, n)   A( x, y, g, j, n)   D( x, y, i, n)   R( x, y, g, j, n)   ph'( x, y, f, g, i, j, k, n)   ps'( x, y, f, g, i, j, k, n)   ch'( x, y, f, g, i, n)

Proof of Theorem bnj580

Step	Hyp	Ref	Expression
1		bnj580.3	. . . . . . 7 |- ( ch <-> ( f Fn n /\ ph /\ ps ) )
2	1	simp1bi 1009	. . . . . 6 |- ( ch -> f Fn n )
3		bnj580.4	. . . . . . . 8 |- ( ph' <-> [. g / f ]. ph )
4		bnj580.5	. . . . . . . 8 |- ( ps' <-> [. g / f ]. ps )
5		bnj580.6	. . . . . . . 8 |- ( ch' <-> [. g / f ]. ch )
6	1, 3, 4, 5	bnj581 34313	. . . . . . 7 |- ( ch' <-> ( g Fn n /\ ph' /\ ps' ) )
7	6	simp1bi 1009	. . . . . 6 |- ( ch' -> g Fn n )
8	2, 7	bnj240 34098	. . . . 5 |- ( ( n e. D /\ ch /\ ch' ) -> ( f Fn n /\ g Fn n ) )
9		bnj580.1	. . . . . . . . . . . . 13 |- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
10		bnj580.2	. . . . . . . . . . . . 13 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
11		bnj580.7	. . . . . . . . . . . . 13 |- D = ( om \ { (/) } )
12	3, 9	bnj154 34283	. . . . . . . . . . . . 13 |- ( ph' <-> ( g ` (/) ) = pred ( x , A , R ) )
13		vex 3037	. . . . . . . . . . . . . 14 |- g e. _V
14	10, 4, 13	bnj540 34297	. . . . . . . . . . . . 13 |- ( ps' <-> A. i e. om ( suc i e. n -> ( g ` suc i ) = U_ y e. ( g ` i ) pred ( y , A , R ) ) )
15		bnj580.8	. . . . . . . . . . . . 13 |- ( th <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
16	15	bnj591 34316	. . . . . . . . . . . . 13 |- ( [. k / j ]. th <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` k ) = ( g ` k ) ) )
17		bnj580.9	. . . . . . . . . . . . 13 |- ( ta <-> A. k e. n ( k _E j -> [. k / j ]. th ) )
18	9, 10, 1, 11, 12, 14, 6, 15, 16, 17	bnj594 34317	. . . . . . . . . . . 12 |- ( ( j e. n /\ ta ) -> th )
19	18	ex 432	. . . . . . . . . . 11 |- ( j e. n -> ( ta -> th ) )
20	19	rgen 2742	. . . . . . . . . 10 |- A. j e. n ( ta -> th )
21		vex 3037	. . . . . . . . . . 11 |- n e. _V
22	21, 17	bnj110 34263	. . . . . . . . . 10 |- ( ( _E Fr n /\ A. j e. n ( ta -> th ) ) -> A. j e. n th )
23	20, 22	mpan2 669	. . . . . . . . 9 |- ( _E Fr n -> A. j e. n th )
24	15	ralbii 2813	. . . . . . . . 9 |- ( A. j e. n th <-> A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
25	23, 24	sylib 196	. . . . . . . 8 |- ( _E Fr n -> A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
26	25	r19.21be 2753	. . . . . . 7 |- A. j e. n ( _E Fr n -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
27	11	bnj923 34173	. . . . . . . . . . . . 13 |- ( n e. D -> n e. om )
28		nnord 6607	. . . . . . . . . . . . 13 |- ( n e. om -> Ord n )
29		ordfr 4807	. . . . . . . . . . . . 13 |- ( Ord n -> _E Fr n )
30	27, 28, 29	3syl 20	. . . . . . . . . . . 12 |- ( n e. D -> _E Fr n )
31	30	3ad2ant1 1015	. . . . . . . . . . 11 |- ( ( n e. D /\ ch /\ ch' ) -> _E Fr n )
32	31	pm4.71ri 631	. . . . . . . . . 10 |- ( ( n e. D /\ ch /\ ch' ) <-> ( _E Fr n /\ ( n e. D /\ ch /\ ch' ) ) )
33	32	imbi1i 323	. . . . . . . . 9 |- ( ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) <-> ( ( _E Fr n /\ ( n e. D /\ ch /\ ch' ) ) -> ( f ` j ) = ( g ` j ) ) )
34		impexp 444	. . . . . . . . 9 |- ( ( ( _E Fr n /\ ( n e. D /\ ch /\ ch' ) ) -> ( f ` j ) = ( g ` j ) ) <-> ( _E Fr n -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) ) )
35	33, 34	bitri 249	. . . . . . . 8 |- ( ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) <-> ( _E Fr n -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) ) )
36	35	ralbii 2813	. . . . . . 7 |- ( A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) <-> A. j e. n ( _E Fr n -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) ) )
37	26, 36	mpbir 209	. . . . . 6 |- A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) )
38		r19.21v 2787	. . . . . 6 |- ( A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) <-> ( ( n e. D /\ ch /\ ch' ) -> A. j e. n ( f ` j ) = ( g ` j ) ) )
39	37, 38	mpbi 208	. . . . 5 |- ( ( n e. D /\ ch /\ ch' ) -> A. j e. n ( f ` j ) = ( g ` j ) )
40		eqfnfv 5883	. . . . . 6 |- ( ( f Fn n /\ g Fn n ) -> ( f = g <-> A. j e. n ( f ` j ) = ( g ` j ) ) )
41	40	biimprd 223	. . . . 5 |- ( ( f Fn n /\ g Fn n ) -> ( A. j e. n ( f ` j ) = ( g ` j ) -> f = g ) )
42	8, 39, 41	sylc 60	. . . 4 |- ( ( n e. D /\ ch /\ ch' ) -> f = g )
43	42	3expib 1197	. . 3 |- ( n e. D -> ( ( ch /\ ch' ) -> f = g ) )
44	43	alrimivv 1728	. 2 |- ( n e. D -> A. f A. g ( ( ch /\ ch' ) -> f = g ) )
45		sbsbc 3256	. . . . . 6 |- ( [ g / f ] ch <-> [. g / f ]. ch )
46	45	anbi2i 692	. . . . 5 |- ( ( ch /\ [ g / f ] ch ) <-> ( ch /\ [. g / f ]. ch ) )
47	46	imbi1i 323	. . . 4 |- ( ( ( ch /\ [ g / f ] ch ) -> f = g ) <-> ( ( ch /\ [. g / f ]. ch ) -> f = g ) )
48	47	2albii 1649	. . 3 |- ( A. f A. g ( ( ch /\ [ g / f ] ch ) -> f = g ) <-> A. f A. g ( ( ch /\ [. g / f ]. ch ) -> f = g ) )
49		nfv 1715	. . . 4 |- F/ g ch
50	49	mo3 2258	. . 3 |- ( E* f ch <-> A. f A. g ( ( ch /\ [ g / f ] ch ) -> f = g ) )
51	5	anbi2i 692	. . . . 5 |- ( ( ch /\ ch' ) <-> ( ch /\ [. g / f ]. ch ) )
52	51	imbi1i 323	. . . 4 |- ( ( ( ch /\ ch' ) -> f = g ) <-> ( ( ch /\ [. g / f ]. ch ) -> f = g ) )
53	52	2albii 1649	. . 3 |- ( A. f A. g ( ( ch /\ ch' ) -> f = g ) <-> A. f A. g ( ( ch /\ [. g / f ]. ch ) -> f = g ) )
54	48, 50, 53	3bitr4i 277	. 2 |- ( E* f ch <-> A. f A. g ( ( ch /\ ch' ) -> f = g ) )
55	44, 54	sylibr 212	1 |- ( n e. D -> E* f ch )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 971   A.wal 1397   = wceq 1399   [wsb 1747   e. wcel 1826   E*wmo 2219   A.wral 2732   [.wsbc 3252   \ cdif 3386   (/)c0 3711   {csn 3944   U_ciun 4243   class class class wbr 4367   _E cep 4703   Fr wfr 4749   Ord word 4791   suc csuc 4794   Fn wfn 5491   ` cfv 5496   omcom 6599   predc-bnj14 34087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-fv 5504  df-om 6600  df-bnj17 34086

This theorem is referenced by:  bnj579  34319