Theorem bnj151 13243

Description: Technical lemma of bnj152 13244. 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).

Hypotheses

Ref	Expression
bnj151.1	|- (ph <-> (f` (/)) = pred(x, A, R))
bnj151.2	|- (ps <-> A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
bnj151.3	|- D = (om \ {(/)})
bnj151.4	|- (th <-> ((R FrSe A /\ x e. A) -> E!f(f Fn n /\ ph /\ ps)))
bnj151.5	|- (ta <-> A.m e. D (m _E n -> [m / n]th))
bnj151.6	|- (ze <-> ((R FrSe A /\ x e. A) -> (f Fn n /\ ph /\ ps)))
bnj151.7	|- (ph' <-> [1o / n]ph)
bnj151.8	|- (ps' <-> [1o / n]ps)
bnj151.9	|- (th' <-> [1o / n]th)
bnj151.10	|- (th0 <-> ((R FrSe A /\ x e. A) -> E.f(f Fn 1o /\ ph' /\ ps')))
bnj151.11	|- (th1 <-> ((R FrSe A /\ x e. A) -> E*f(f Fn 1o /\ ph' /\ ps')))
bnj151.12	|- (ze' <-> [1o / n]ze)
bnj151.13	|- F = {<.(/), pred(x, A, R)>.}
bnj151.14	|- (ph" <-> [F / f]ph')
bnj151.15	|- (ps" <-> [F / f]ps')
bnj151.16	|- (ze" <-> [F / f]ze')
bnj151.17	|- (ze0 <-> (f Fn 1o /\ ph' /\ ps'))
bnj151.18	|- (ze1 <-> [g / f]ze0)
bnj151.19	|- (ph1 <-> [g / f]ph')
bnj151.20	|- (ps1 <-> [g / f]ps')

Assertion

Ref	Expression
bnj151	|- (n = 1o -> ((n e. D /\ ta) -> th))

Distinct variable groups:   A,f,g,x   A,n,f,x   f,F,i,y   R,f,g,x   R,n   f,ze1   f,ze"   g,ze0   i,n,y   m,n

Proof of Theorem bnj151

Step	Hyp	Ref	Expression
1		eu5 1805	. . . . 5 |- (E!f(f Fn 1o /\ ph' /\ ps') <-> (E.f(f Fn 1o /\ ph' /\ ps') /\ E*f(f Fn 1o /\ ph' /\ ps')))
2		bnj151.1	. . . . . . 7 |- (ph <-> (f` (/)) = pred(x, A, R))
3		bnj151.2	. . . . . . 7 |- (ps <-> A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
4		bnj151.6	. . . . . . 7 |- (ze <-> ((R FrSe A /\ x e. A) -> (f Fn n /\ ph /\ ps)))
5		bnj151.7	. . . . . . 7 |- (ph' <-> [1o / n]ph)
6		bnj151.8	. . . . . . 7 |- (ps' <-> [1o / n]ps)
7		bnj151.10	. . . . . . 7 |- (th0 <-> ((R FrSe A /\ x e. A) -> E.f(f Fn 1o /\ ph' /\ ps')))
8		bnj151.12	. . . . . . 7 |- (ze' <-> [1o / n]ze)
9		bnj151.13	. . . . . . 7 |- F = {<.(/), pred(x, A, R)>.}
10		bnj151.14	. . . . . . 7 |- (ph" <-> [F / f]ph')
11		bnj151.15	. . . . . . 7 |- (ps" <-> [F / f]ps')
12		bnj151.16	. . . . . . 7 |- (ze" <-> [F / f]ze')
13	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	bnj150 13242	. . . . . 6 |- th0
14	13, 7	mpbi 206	. . . . 5 |- ((R FrSe A /\ x e. A) -> E.f(f Fn 1o /\ ph' /\ ps'))
15		bnj151.11	. . . . . . 7 |- (th1 <-> ((R FrSe A /\ x e. A) -> E*f(f Fn 1o /\ ph' /\ ps')))
16		bnj151.17	. . . . . . 7 |- (ze0 <-> (f Fn 1o /\ ph' /\ ps'))
17		bnj151.18	. . . . . . 7 |- (ze1 <-> [g / f]ze0)
18		bnj151.19	. . . . . . 7 |- (ph1 <-> [g / f]ph')
19		bnj151.20	. . . . . . 7 |- (ps1 <-> [g / f]ps')
20	2, 5	bnj118 13228	. . . . . . 7 |- (ph' <-> (f` (/)) = pred(x, A, R))
21	15, 16, 17, 18, 19, 20	bnj149 13241	. . . . . 6 |- th1
22	21, 15	mpbi 206	. . . . 5 |- ((R FrSe A /\ x e. A) -> E*f(f Fn 1o /\ ph' /\ ps'))
23	1, 14, 22	sylanbrc 527	. . . 4 |- ((R FrSe A /\ x e. A) -> E!f(f Fn 1o /\ ph' /\ ps'))
24		bnj151.4	. . . . 5 |- (th <-> ((R FrSe A /\ x e. A) -> E!f(f Fn n /\ ph /\ ps)))
25		bnj151.9	. . . . 5 |- (th' <-> [1o / n]th)
26	24, 5, 6, 25	bnj130 13240	. . . 4 |- (th' <-> ((R FrSe A /\ x e. A) -> E!f(f Fn 1o /\ ph' /\ ps')))
27	23, 26	mpbir 207	. . 3 |- th'
28	25	bnj117 12461	. . 3 |- (n = 1o -> (th <-> th'))
29	27, 28	mpbiri 211	. 2 |- (n = 1o -> th)
30	29	a1d 15	1 |- (n = 1o -> ((n e. D /\ ta) -> th))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  E!weu 1771  E*wmo 1772  A.wral 2105   \ cdif 2590  (/)c0 2875  {csn 3044  <.cop 3046  U_ciun 3255   class class class wbr 3338   _E cep 3581  suc csuc 3659  omcom 3949   Fn wfn 3993  ` cfv 3998  1oc1o 5172   predsyn-bnj14 12023   FrSe syn-bnj15 12027

This theorem is referenced by:  bnj152 13244  bnj153 13247

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-1o 5177  df-bnj13 12026  df-bnj15 12028

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