Theorem bnj607 29723

Description: Technical lemma for bnj852 29728. 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
bnj607.5	|- ( th <-> A. m e. D ( m _E n -> [. m / n ]. ch ) )
bnj607.13	|- ( ph" <-> [. G / f ]. ph )
bnj607.14	|- ( ps" <-> [. G / f ]. ps )
bnj607.17	|- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
bnj607.19	|- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
bnj607.28	|- G e. _V
bnj607.31	|- ( ch' <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn m /\ ph' /\ ps' ) ) )
bnj607.32	|- ( ph" <-> ( G ` (/) ) = pred ( x , A , R ) )
bnj607.33	|- ( ps" <-> A. i e. om ( suc i e. n -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
bnj607.37	|- ( ( n =/= 1o /\ n e. D ) -> E. m E. p et )
bnj607.38	|- ( ( th /\ m e. D /\ m _E n ) -> ch' )
bnj607.41	|- ( ( R FrSe A /\ ta /\ et ) -> G Fn n )
bnj607.42	|- ( ( R FrSe A /\ ta /\ et ) -> ph" )
bnj607.43	|- ( ( R FrSe A /\ ta /\ et ) -> ps" )
bnj607.1	|- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
bnj607.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj607.400	|- ( ph0 <-> [. h / f ]. ph )
bnj607.401	|- ( ps0 <-> [. h / f ]. ps )
bnj607.300	|- ( ph1 <-> [. G / h ]. ph0 )
bnj607.301	|- ( ps1 <-> [. G / h ]. ps0 )

Assertion

Ref	Expression
bnj607	|- ( ( n =/= 1o /\ n e. D /\ th ) -> ( ( R FrSe A /\ x e. A ) -> E. f ( f Fn n /\ ph /\ ps ) ) )

Distinct variable groups:   A, f, h   A, m, f   A, p, f   h, G, i, y   R, f, h   R, m   R, p   et, f   f, i, y   f, n, h   x, f, h   ph, h   ps, h   m, n   ph, m   ps, m   x, m   n, p   ph, p   ps, p   th, p   x, p

Allowed substitution hints:   ph( x, y, f, i, n)   ps( x, y, f, i, n)   ch( x, y, f, h, i, m, n, p)   th( x, y, f, h, i, m, n)   ta( x, y, f, h, i, m, n, p)   et( x, y, h, i, m, n, p)   A( x, y, i, n)   D( x, y, f, h, i, m, n, p)   R( x, y, i, n)   G( x, f, m, n, p)   ph'( x, y, f, h, i, m, n, p)   ps'( x, y, f, h, i, m, n, p)   ch'( x, y, f, h, i, m, n, p)   ph"( x, y, f, h, i, m, n, p)   ps"( x, y, f, h, i, m, n, p)   ph0( x, y, f, h, i, m, n, p)   ps0( x, y, f, h, i, m, n, p)   ph1( x, y, f, h, i, m, n, p)   ps1( x, y, f, h, i, m, n, p)

Proof of Theorem bnj607

Step	Hyp	Ref	Expression
1		bnj607.37	. . . . 5 |- ( ( n =/= 1o /\ n e. D ) -> E. m E. p et )
2	1	anim1i 570	. . . 4 |- ( ( ( n =/= 1o /\ n e. D ) /\ th ) -> ( E. m E. p et /\ th ) )
3		nfv 1751	. . . . . . 7 |- F/ p th
4	3	19.41 2026	. . . . . 6 |- ( E. p ( et /\ th ) <-> ( E. p et /\ th ) )
5	4	exbii 1712	. . . . 5 |- ( E. m E. p ( et /\ th ) <-> E. m ( E. p et /\ th ) )
6		bnj607.5	. . . . . . . 8 |- ( th <-> A. m e. D ( m _E n -> [. m / n ]. ch ) )
7	6	bnj1095 29589	. . . . . . 7 |- ( th -> A. m th )
8	7	nfi 1670	. . . . . 6 |- F/ m th
9	8	19.41 2026	. . . . 5 |- ( E. m ( E. p et /\ th ) <-> ( E. m E. p et /\ th ) )
10	5, 9	bitr2i 253	. . . 4 |- ( ( E. m E. p et /\ th ) <-> E. m E. p ( et /\ th ) )
11	2, 10	sylib 199	. . 3 |- ( ( ( n =/= 1o /\ n e. D ) /\ th ) -> E. m E. p ( et /\ th ) )
12		bnj607.19	. . . . . . . . . 10 |- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
13	12	bnj1232 29611	. . . . . . . . 9 |- ( et -> m e. D )
14		bnj219 29537	. . . . . . . . . 10 |- ( n = suc m -> m _E n )
15	12, 14	bnj770 29570	. . . . . . . . 9 |- ( et -> m _E n )
16	13, 15	jca 534	. . . . . . . 8 |- ( et -> ( m e. D /\ m _E n ) )
17	16	anim1i 570	. . . . . . 7 |- ( ( et /\ th ) -> ( ( m e. D /\ m _E n ) /\ th ) )
18		bnj170 29499	. . . . . . 7 |- ( ( th /\ m e. D /\ m _E n ) <-> ( ( m e. D /\ m _E n ) /\ th ) )
19	17, 18	sylibr 215	. . . . . 6 |- ( ( et /\ th ) -> ( th /\ m e. D /\ m _E n ) )
20		bnj607.38	. . . . . 6 |- ( ( th /\ m e. D /\ m _E n ) -> ch' )
21	19, 20	syl 17	. . . . 5 |- ( ( et /\ th ) -> ch' )
22		simpl 458	. . . . 5 |- ( ( et /\ th ) -> et )
23	21, 22	jca 534	. . . 4 |- ( ( et /\ th ) -> ( ch' /\ et ) )
24	23	2eximi 1703	. . 3 |- ( E. m E. p ( et /\ th ) -> E. m E. p ( ch' /\ et ) )
25		bnj607.31	. . . . . . . . . . . 12 |- ( ch' <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn m /\ ph' /\ ps' ) ) )
26	25	biimpi 197	. . . . . . . . . . 11 |- ( ch' -> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn m /\ ph' /\ ps' ) ) )
27		euex 2290	. . . . . . . . . . 11 |- ( E! f ( f Fn m /\ ph' /\ ps' ) -> E. f ( f Fn m /\ ph' /\ ps' ) )
28	26, 27	syl6 34	. . . . . . . . . 10 |- ( ch' -> ( ( R FrSe A /\ x e. A ) -> E. f ( f Fn m /\ ph' /\ ps' ) ) )
29	28	impcom 431	. . . . . . . . 9 |- ( ( ( R FrSe A /\ x e. A ) /\ ch' ) -> E. f ( f Fn m /\ ph' /\ ps' ) )
30		bnj607.17	. . . . . . . . 9 |- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
31	29, 30	bnj1198 29603	. . . . . . . 8 |- ( ( ( R FrSe A /\ x e. A ) /\ ch' ) -> E. f ta )
32	31	adantrr 721	. . . . . . 7 |- ( ( ( R FrSe A /\ x e. A ) /\ ( ch' /\ et ) ) -> E. f ta )
33		id 23	. . . . . . . . . . 11 |- ( ( R FrSe A /\ ta /\ et ) -> ( R FrSe A /\ ta /\ et ) )
34	33	3com23 1211	. . . . . . . . . 10 |- ( ( R FrSe A /\ et /\ ta ) -> ( R FrSe A /\ ta /\ et ) )
35	34	3expia 1207	. . . . . . . . 9 |- ( ( R FrSe A /\ et ) -> ( ta -> ( R FrSe A /\ ta /\ et ) ) )
36	35	eximdv 1754	. . . . . . . 8 |- ( ( R FrSe A /\ et ) -> ( E. f ta -> E. f ( R FrSe A /\ ta /\ et ) ) )
37	36	ad2ant2rl 753	. . . . . . 7 |- ( ( ( R FrSe A /\ x e. A ) /\ ( ch' /\ et ) ) -> ( E. f ta -> E. f ( R FrSe A /\ ta /\ et ) ) )
38	32, 37	mpd 15	. . . . . 6 |- ( ( ( R FrSe A /\ x e. A ) /\ ( ch' /\ et ) ) -> E. f ( R FrSe A /\ ta /\ et ) )
39		bnj607.41	. . . . . . . 8 |- ( ( R FrSe A /\ ta /\ et ) -> G Fn n )
40		bnj607.42	. . . . . . . 8 |- ( ( R FrSe A /\ ta /\ et ) -> ph" )
41		bnj607.43	. . . . . . . 8 |- ( ( R FrSe A /\ ta /\ et ) -> ps" )
42	39, 40, 41	3jca 1185	. . . . . . 7 |- ( ( R FrSe A /\ ta /\ et ) -> ( G Fn n /\ ph" /\ ps" ) )
43	42	eximi 1702	. . . . . 6 |- ( E. f ( R FrSe A /\ ta /\ et ) -> E. f ( G Fn n /\ ph" /\ ps" ) )
44		nfe1 1890	. . . . . . 7 |- F/ f E. f ( f Fn n /\ ph /\ ps )
45		bnj607.28	. . . . . . . . 9 |- G e. _V
46		nfcv 2584	. . . . . . . . . 10 |- F/_ h G
47		nfv 1751	. . . . . . . . . . 11 |- F/ h G Fn n
48		bnj607.300	. . . . . . . . . . . 12 |- ( ph1 <-> [. G / h ]. ph0 )
49		nfsbc1v 3319	. . . . . . . . . . . 12 |- F/ h [. G / h ]. ph0
50	48, 49	nfxfr 1692	. . . . . . . . . . 11 |- F/ h ph1
51		bnj607.301	. . . . . . . . . . . 12 |- ( ps1 <-> [. G / h ]. ps0 )
52		nfsbc1v 3319	. . . . . . . . . . . 12 |- F/ h [. G / h ]. ps0
53	51, 52	nfxfr 1692	. . . . . . . . . . 11 |- F/ h ps1
54	47, 50, 53	nf3an 1986	. . . . . . . . . 10 |- F/ h ( G Fn n /\ ph1 /\ ps1 )
55		fneq1 5679	. . . . . . . . . . 11 |- ( h = G -> ( h Fn n <-> G Fn n ) )
56		sbceq1a 3310	. . . . . . . . . . . 12 |- ( h = G -> ( ph0 <-> [. G / h ]. ph0 ) )
57	56, 48	syl6bbr 266	. . . . . . . . . . 11 |- ( h = G -> ( ph0 <-> ph1 ) )
58		sbceq1a 3310	. . . . . . . . . . . 12 |- ( h = G -> ( ps0 <-> [. G / h ]. ps0 ) )
59	58, 51	syl6bbr 266	. . . . . . . . . . 11 |- ( h = G -> ( ps0 <-> ps1 ) )
60	55, 57, 59	3anbi123d 1335	. . . . . . . . . 10 |- ( h = G -> ( ( h Fn n /\ ph0 /\ ps0 ) <-> ( G Fn n /\ ph1 /\ ps1 ) ) )
61	46, 54, 60	spcegf 3162	. . . . . . . . 9 |- ( G e. _V -> ( ( G Fn n /\ ph1 /\ ps1 ) -> E. h ( h Fn n /\ ph0 /\ ps0 ) ) )
62	45, 61	ax-mp 5	. . . . . . . 8 |- ( ( G Fn n /\ ph1 /\ ps1 ) -> E. h ( h Fn n /\ ph0 /\ ps0 ) )
63		bnj607.32	. . . . . . . . . . . 12 |- ( ph" <-> ( G ` (/) ) = pred ( x , A , R ) )
64		bnj607.400	. . . . . . . . . . . . . 14 |- ( ph0 <-> [. h / f ]. ph )
65		bnj607.1	. . . . . . . . . . . . . 14 |- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
66	64, 65	bnj154 29685	. . . . . . . . . . . . 13 |- ( ph0 <-> ( h ` (/) ) = pred ( x , A , R ) )
67	66, 48, 45	bnj526 29695	. . . . . . . . . . . 12 |- ( ph1 <-> ( G ` (/) ) = pred ( x , A , R ) )
68	63, 67	bitr4i 255	. . . . . . . . . . 11 |- ( ph" <-> ph1 )
69		bnj607.33	. . . . . . . . . . . 12 |- ( ps" <-> A. i e. om ( suc i e. n -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
70		bnj607.2	. . . . . . . . . . . . . 14 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
71		bnj607.401	. . . . . . . . . . . . . 14 |- ( ps0 <-> [. h / f ]. ps )
72		vex 3084	. . . . . . . . . . . . . 14 |- h e. _V
73	70, 71, 72	bnj540 29699	. . . . . . . . . . . . 13 |- ( ps0 <-> A. i e. om ( suc i e. n -> ( h ` suc i ) = U_ y e. ( h ` i ) pred ( y , A , R ) ) )
74	73, 51, 45	bnj540 29699	. . . . . . . . . . . 12 |- ( ps1 <-> A. i e. om ( suc i e. n -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
75	69, 74	bitr4i 255	. . . . . . . . . . 11 |- ( ps" <-> ps1 )
76	68, 75	anbi12i 701	. . . . . . . . . 10 |- ( ( ph" /\ ps" ) <-> ( ph1 /\ ps1 ) )
77	76	anbi2i 698	. . . . . . . . 9 |- ( ( G Fn n /\ ( ph" /\ ps" ) ) <-> ( G Fn n /\ ( ph1 /\ ps1 ) ) )
78		3anass 986	. . . . . . . . 9 |- ( ( G Fn n /\ ph" /\ ps" ) <-> ( G Fn n /\ ( ph" /\ ps" ) ) )
79		3anass 986	. . . . . . . . 9 |- ( ( G Fn n /\ ph1 /\ ps1 ) <-> ( G Fn n /\ ( ph1 /\ ps1 ) ) )
80	77, 78, 79	3bitr4i 280	. . . . . . . 8 |- ( ( G Fn n /\ ph" /\ ps" ) <-> ( G Fn n /\ ph1 /\ ps1 ) )
81		nfv 1751	. . . . . . . . 9 |- F/ h ( f Fn n /\ ph /\ ps )
82		nfv 1751	. . . . . . . . . 10 |- F/ f h Fn n
83		nfsbc1v 3319	. . . . . . . . . . 11 |- F/ f [. h / f ]. ph
84	64, 83	nfxfr 1692	. . . . . . . . . 10 |- F/ f ph0
85		nfsbc1v 3319	. . . . . . . . . . 11 |- F/ f [. h / f ]. ps
86	71, 85	nfxfr 1692	. . . . . . . . . 10 |- F/ f ps0
87	82, 84, 86	nf3an 1986	. . . . . . . . 9 |- F/ f ( h Fn n /\ ph0 /\ ps0 )
88		fneq1 5679	. . . . . . . . . 10 |- ( f = h -> ( f Fn n <-> h Fn n ) )
89		sbceq1a 3310	. . . . . . . . . . 11 |- ( f = h -> ( ph <-> [. h / f ]. ph ) )
90	89, 64	syl6bbr 266	. . . . . . . . . 10 |- ( f = h -> ( ph <-> ph0 ) )
91		sbceq1a 3310	. . . . . . . . . . 11 |- ( f = h -> ( ps <-> [. h / f ]. ps ) )
92	91, 71	syl6bbr 266	. . . . . . . . . 10 |- ( f = h -> ( ps <-> ps0 ) )
93	88, 90, 92	3anbi123d 1335	. . . . . . . . 9 |- ( f = h -> ( ( f Fn n /\ ph /\ ps ) <-> ( h Fn n /\ ph0 /\ ps0 ) ) )
94	81, 87, 93	cbvex 2076	. . . . . . . 8 |- ( E. f ( f Fn n /\ ph /\ ps ) <-> E. h ( h Fn n /\ ph0 /\ ps0 ) )
95	62, 80, 94	3imtr4i 269	. . . . . . 7 |- ( ( G Fn n /\ ph" /\ ps" ) -> E. f ( f Fn n /\ ph /\ ps ) )
96	44, 95	exlimi 1968	. . . . . 6 |- ( E. f ( G Fn n /\ ph" /\ ps" ) -> E. f ( f Fn n /\ ph /\ ps ) )
97	38, 43, 96	3syl 18	. . . . 5 |- ( ( ( R FrSe A /\ x e. A ) /\ ( ch' /\ et ) ) -> E. f ( f Fn n /\ ph /\ ps ) )
98	97	expcom 436	. . . 4 |- ( ( ch' /\ et ) -> ( ( R FrSe A /\ x e. A ) -> E. f ( f Fn n /\ ph /\ ps ) ) )
99	98	exlimivv 1767	. . 3 |- ( E. m E. p ( ch' /\ et ) -> ( ( R FrSe A /\ x e. A ) -> E. f ( f Fn n /\ ph /\ ps ) ) )
100	11, 24, 99	3syl 18	. 2 |- ( ( ( n =/= 1o /\ n e. D ) /\ th ) -> ( ( R FrSe A /\ x e. A ) -> E. f ( f Fn n /\ ph /\ ps ) ) )
101	100	3impa 1200	1 |- ( ( n =/= 1o /\ n e. D /\ th ) -> ( ( R FrSe A /\ x e. A ) -> E. f ( f Fn n /\ ph /\ ps ) ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   E.wex 1659   e. wcel 1868   E!weu 2265   =/= wne 2618   A.wral 2775   _Vcvv 3081   [.wsbc 3299   (/)c0 3761   U_ciun 4296   class class class wbr 4420   _E cep 4759   suc csuc 5441   Fn wfn 5593   ` cfv 5598   omcom 6703   1oc1o 7180   /\ w-bnj17 29487   predc-bnj14 29489   FrSe w-bnj15 29493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-eprel 4761  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-fv 5606  df-bnj17 29488

This theorem is referenced by:  bnj600  29726