Theorem bnj908 13329

Description: Technical lemma of bnj69 13455. 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
bnj908.1	|- (ph <-> (f` (/)) = pred(x, A, R))
bnj908.2	|- (ps <-> A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
bnj908.3	|- D = (om \ {(/)})
bnj908.4	|- (ch <-> ((R FrSe A /\ x e. A) -> E!f(f Fn n /\ ph /\ ps)))
bnj908.5	|- (th <-> A.m e. D (m _E n -> [m / n]ch))
bnj908.10	|- (ph' <-> [m / n]ph)
bnj908.11	|- (ps' <-> [m / n]ps)
bnj908.12	|- (ch' <-> [m / n]ch)
bnj908.13	|- (ph" <-> [G / f]ph)
bnj908.14	|- (ps" <-> [G / f]ps)
bnj908.15	|- (ch" <-> [G / f]ch)
bnj908.16	|- G = (f u. {<.m, U_y e. (f` p) pred(y, A, R)>.})
bnj908.17	|- (ta <-> (f Fn m /\ ph' /\ ps'))
bnj908.18	|- (si <-> (m e. D /\ n = suc m /\ p e. m))
bnj908.19	|- (et <-> (m e. D /\ n = suc m /\ p e. om /\ m = suc p))
bnj908.20	|- (ze <-> (i e. om /\ suc i e. n /\ m = suc i))
bnj908.21	|- (rh <-> (i e. om /\ suc i e. n /\ m =/= suc i))
bnj908.22	|- B = U_y e. (f` i) pred(y, A, R)
bnj908.23	|- C = U_y e. (f` p) pred(y, A, R)
bnj908.24	|- K = U_y e. (G` i) pred(y, A, R)
bnj908.25	|- L = U_y e. (G` p) pred(y, A, R)
bnj908.26	|- G = (f u. {<.m, C>.})

Assertion

Ref	Expression
bnj908	|- ((R FrSe A /\ x e. A /\ ch' /\ et) -> E.f(G Fn n /\ ph" /\ ps"))

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

Proof of Theorem bnj908

Step	Hyp	Ref	Expression
1		19.41v 1685	. . . 4 |- (E.f((ta /\ et) /\ R FrSe A) <-> (E.f(ta /\ et) /\ R FrSe A))
2		19.41v 1685	. . . . 5 |- (E.f(ta /\ et) <-> (E.fta /\ et))
3		bnj248 12087	. . . . . 6 |- ((R FrSe A /\ x e. A /\ ch' /\ et) <-> (((R FrSe A /\ x e. A) /\ ch') /\ et))
4		bnj908.4	. . . . . . . . . . 11 |- (ch <-> ((R FrSe A /\ x e. A) -> E!f(f Fn n /\ ph /\ ps)))
5		bnj908.10	. . . . . . . . . . 11 |- (ph' <-> [m / n]ph)
6		bnj908.11	. . . . . . . . . . 11 |- (ps' <-> [m / n]ps)
7		bnj908.12	. . . . . . . . . . 11 |- (ch' <-> [m / n]ch)
8		visset 2295	. . . . . . . . . . 11 |- m e. _V
9	4, 5, 6, 7, 8	bnj207 13248	. . . . . . . . . 10 |- (ch' <-> ((R FrSe A /\ x e. A) -> E!f(f Fn m /\ ph' /\ ps')))
10	9	biimpi 168	. . . . . . . . 9 |- (ch' -> ((R FrSe A /\ x e. A) -> E!f(f Fn m /\ ph' /\ ps')))
11		euex 1788	. . . . . . . . 9 |- (E!f(f Fn m /\ ph' /\ ps') -> E.f(f Fn m /\ ph' /\ ps'))
12	10, 11	syl6 25	. . . . . . . 8 |- (ch' -> ((R FrSe A /\ x e. A) -> E.f(f Fn m /\ ph' /\ ps')))
13	12	impcom 378	. . . . . . 7 |- (((R FrSe A /\ x e. A) /\ ch') -> E.f(f Fn m /\ ph' /\ ps'))
14		bnj908.17	. . . . . . 7 |- (ta <-> (f Fn m /\ ph' /\ ps'))
15	13, 14	bnj1198 12974	. . . . . 6 |- (((R FrSe A /\ x e. A) /\ ch') -> E.fta)
16	3, 15	bnj832 12706	. . . . 5 |- ((R FrSe A /\ x e. A /\ ch' /\ et) -> E.fta)
17		bnj645 12580	. . . . 5 |- ((R FrSe A /\ x e. A /\ ch' /\ et) -> et)
18	2, 16, 17	sylanbrc 527	. . . 4 |- ((R FrSe A /\ x e. A /\ ch' /\ et) -> E.f(ta /\ et))
19		bnj642 12577	. . . 4 |- ((R FrSe A /\ x e. A /\ ch' /\ et) -> R FrSe A)
20	1, 18, 19	sylanbrc 527	. . 3 |- ((R FrSe A /\ x e. A /\ ch' /\ et) -> E.f((ta /\ et) /\ R FrSe A))
21		bnj170 12034	. . 3 |- ((R FrSe A /\ ta /\ et) <-> ((ta /\ et) /\ R FrSe A))
22	20, 21	bnj1198 12974	. 2 |- ((R FrSe A /\ x e. A /\ ch' /\ et) -> E.f(R FrSe A /\ ta /\ et))
23		bnj908.18	. . . 4 |- (si <-> (m e. D /\ n = suc m /\ p e. m))
24		bnj908.19	. . . 4 |- (et <-> (m e. D /\ n = suc m /\ p e. om /\ m = suc p))
25		bnj908.1	. . . . . 6 |- (ph <-> (f` (/)) = pred(x, A, R))
26	25, 5, 8	bnj523 13262	. . . . 5 |- (ph' <-> (f` (/)) = pred(x, A, R))
27		bnj908.2	. . . . . 6 |- (ps <-> A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
28	27, 6, 8	bnj539 13266	. . . . 5 |- (ps' <-> A.i e. om (suc i e. m -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
29		bnj908.3	. . . . 5 |- D = (om \ {(/)})
30		bnj908.16	. . . . 5 |- G = (f u. {<.m, U_y e. (f` p) pred(y, A, R)>.})
31	26, 28, 29, 30, 14, 23	bnj544 13270	. . . 4 |- ((R FrSe A /\ ta /\ si) -> G Fn n)
32	23, 24, 31	bnj561 13283	. . 3 |- ((R FrSe A /\ ta /\ et) -> G Fn n)
33		bnj908.13	. . . . . 6 |- (ph" <-> [G / f]ph)
34	30	bnj528 13264	. . . . . 6 |- G e. _V
35	25, 33, 34	bnj609 13306	. . . . 5 |- (ph" <-> (G` (/)) = pred(x, A, R))
36	26, 29, 30, 14, 23, 31, 35	bnj545 13271	. . . 4 |- ((R FrSe A /\ ta /\ si) -> ph")
37	23, 24, 36	bnj562 13284	. . 3 |- ((R FrSe A /\ ta /\ et) -> ph")
38		bnj908.20	. . . 4 |- (ze <-> (i e. om /\ suc i e. n /\ m = suc i))
39		bnj908.22	. . . 4 |- B = U_y e. (f` i) pred(y, A, R)
40		bnj908.23	. . . 4 |- C = U_y e. (f` p) pred(y, A, R)
41		bnj908.24	. . . 4 |- K = U_y e. (G` i) pred(y, A, R)
42		bnj908.25	. . . 4 |- L = U_y e. (G` p) pred(y, A, R)
43		bnj908.26	. . . 4 |- G = (f u. {<.m, C>.})
44		bnj908.21	. . . 4 |- (rh <-> (i e. om /\ suc i e. n /\ m =/= suc i))
45		bnj908.14	. . . . 5 |- (ps" <-> [G / f]ps)
46	27, 45, 34	bnj611 13307	. . . 4 |- (ps" <-> A.i e. om (suc i e. n -> (G` suc i) = U_y e. (G` i) pred(y, A, R)))
47	29, 30, 14, 23, 24, 38, 39, 40, 41, 42, 43, 26, 28, 31, 44, 32, 46	bnj571 13289	. . 3 |- ((R FrSe A /\ ta /\ et) -> ps")
48	32, 37, 47	3jca 1050	. 2 |- ((R FrSe A /\ ta /\ et) -> (G Fn n /\ ph" /\ ps"))
49	22, 48	bnj593 12556	1 |- ((R FrSe A /\ x e. A /\ ch' /\ et) -> E.f(G Fn n /\ ph" /\ ps"))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  E!weu 1771   =/= wne 2017  A.wral 2105   \ cdif 2590   u. cun 2591  (/)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   /\ syn-bnj17 12019   predsyn-bnj14 12023   FrSe syn-bnj15 12027

This theorem is referenced by:  bnj909 13330

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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  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-nel 2020  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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-fv 4014  df-bnj17 12020  df-bnj14 12024  df-bnj13 12026  df-bnj15 12028

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