Theorem bnj580 28990

Description: Technical lemma for bnj579 28991. 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 972	. . . . . 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 28985	. . . . . . 7 |- ( ch' <-> ( g Fn n /\ ph' /\ ps' ) )
7	6	simp1bi 972	. . . . . 6 |- ( ch' -> g Fn n )
8	2, 7	bnj240 28769	. . . . 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 28955	. . . . . . . . . . . . 13 |- ( ph' <-> ( g ` (/) ) = pred ( x , A , R ) )
13		vex 2919	. . . . . . . . . . . . . 14 |- g e. _V
14	10, 4, 13	bnj540 28969	. . . . . . . . . . . . 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 28988	. . . . . . . . . . . . 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 28989	. . . . . . . . . . . 12 |- ( ( j e. n /\ ta ) -> th )
19	18	ex 424	. . . . . . . . . . 11 |- ( j e. n -> ( ta -> th ) )
20	19	rgen 2731	. . . . . . . . . 10 |- A. j e. n ( ta -> th )
21		vex 2919	. . . . . . . . . . 11 |- n e. _V
22	21, 17	bnj110 28935	. . . . . . . . . 10 |- ( ( _E Fr n /\ A. j e. n ( ta -> th ) ) -> A. j e. n th )
23	20, 22	mpan2 653	. . . . . . . . 9 |- ( _E Fr n -> A. j e. n th )
24	15	ralbii 2690	. . . . . . . . 9 |- ( A. j e. n th <-> A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
25	23, 24	sylib 189	. . . . . . . 8 |- ( _E Fr n -> A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
26	25	r19.21be 2767	. . . . . . 7 |- A. j e. n ( _E Fr n -> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
27	11	bnj923 28843	. . . . . . . . . . . . 13 |- ( n e. D -> n e. om )
28		nnord 4812	. . . . . . . . . . . . 13 |- ( n e. om -> Ord n )
29		ordfr 4556	. . . . . . . . . . . . 13 |- ( Ord n -> _E Fr n )
30	27, 28, 29	3syl 19	. . . . . . . . . . . 12 |- ( n e. D -> _E Fr n )
31	30	3ad2ant1 978	. . . . . . . . . . 11 |- ( ( n e. D /\ ch /\ ch' ) -> _E Fr n )
32	31	pm4.71ri 615	. . . . . . . . . 10 |- ( ( n e. D /\ ch /\ ch' ) <-> ( _E Fr n /\ ( n e. D /\ ch /\ ch' ) ) )
33	32	imbi1i 316	. . . . . . . . 9 |- ( ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) <-> ( ( _E Fr n /\ ( n e. D /\ ch /\ ch' ) ) -> ( f ` j ) = ( g ` j ) ) )
34		impexp 434	. . . . . . . . 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 241	. . . . . . . 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 2690	. . . . . . 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 201	. . . . . 6 |- A. j e. n ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) )
38		r19.21v 2753	. . . . . 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 200	. . . . 5 |- ( ( n e. D /\ ch /\ ch' ) -> A. j e. n ( f ` j ) = ( g ` j ) )
40		eqfnfv 5786	. . . . . 6 |- ( ( f Fn n /\ g Fn n ) -> ( f = g <-> A. j e. n ( f ` j ) = ( g ` j ) ) )
41	40	biimprd 215	. . . . 5 |- ( ( f Fn n /\ g Fn n ) -> ( A. j e. n ( f ` j ) = ( g ` j ) -> f = g ) )
42	8, 39, 41	sylc 58	. . . 4 |- ( ( n e. D /\ ch /\ ch' ) -> f = g )
43	42	3expib 1156	. . 3 |- ( n e. D -> ( ( ch /\ ch' ) -> f = g ) )
44	43	alrimivv 1639	. 2 |- ( n e. D -> A. f A. g ( ( ch /\ ch' ) -> f = g ) )
45		sbsbc 3125	. . . . . 6 |- ( [ g / f ] ch <-> [. g / f ]. ch )
46	45	anbi2i 676	. . . . 5 |- ( ( ch /\ [ g / f ] ch ) <-> ( ch /\ [. g / f ]. ch ) )
47	46	imbi1i 316	. . . 4 |- ( ( ( ch /\ [ g / f ] ch ) -> f = g ) <-> ( ( ch /\ [. g / f ]. ch ) -> f = g ) )
48	47	2albii 1573	. . 3 |- ( A. f A. g ( ( ch /\ [ g / f ] ch ) -> f = g ) <-> A. f A. g ( ( ch /\ [. g / f ]. ch ) -> f = g ) )
49		nfv 1626	. . . 4 |- F/ g ch
50	49	mo3 2285	. . 3 |- ( E* f ch <-> A. f A. g ( ( ch /\ [ g / f ] ch ) -> f = g ) )
51	5	anbi2i 676	. . . . 5 |- ( ( ch /\ ch' ) <-> ( ch /\ [. g / f ]. ch ) )
52	51	imbi1i 316	. . . 4 |- ( ( ( ch /\ ch' ) -> f = g ) <-> ( ( ch /\ [. g / f ]. ch ) -> f = g ) )
53	52	2albii 1573	. . 3 |- ( A. f A. g ( ( ch /\ ch' ) -> f = g ) <-> A. f A. g ( ( ch /\ [. g / f ]. ch ) -> f = g ) )
54	48, 50, 53	3bitr4i 269	. 2 |- ( E* f ch <-> A. f A. g ( ( ch /\ ch' ) -> f = g ) )
55	44, 54	sylibr 204	1 |- ( n e. D -> E* f ch )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   /\ w3a 936   A.wal 1546   = wceq 1649   [wsb 1655   e. wcel 1721   E*wmo 2255   A.wral 2666   [.wsbc 3121   \ cdif 3277   (/)c0 3588   {csn 3774   U_ciun 4053   class class class wbr 4172   _E cep 4452   Fr wfr 4498   Ord word 4540   suc csuc 4543   omcom 4804   Fn wfn 5408   ` cfv 5413   predc-bnj14 28758

This theorem is referenced by:  bnj579  28991

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421  df-bnj17 28757