Theorem bnj1016 13376

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
bnj1016.1	|- (ph <-> (f` (/)) = pred(X, A, R))
bnj1016.2	|- (ps <-> A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
bnj1016.3	|- (ch <-> (n e. D /\ f Fn n /\ ph /\ ps))
bnj1016.4	|- (th <-> (R FrSe A /\ X e. A /\ y e. trCl(X, A, R) /\ z e. pred(y, A, R)))
bnj1016.5	|- (ta <-> (m e. om /\ n = suc m /\ p = suc n))
bnj1016.7	|- (ph' <-> [p / n]ph)
bnj1016.8	|- (ps' <-> [p / n]ps)
bnj1016.9	|- (ch' <-> [p / n]ch)
bnj1016.10	|- (ph" <-> [G / f]ph')
bnj1016.11	|- (ps" <-> [G / f]ps')
bnj1016.12	|- (ch" <-> [G / f]ch')
bnj1016.13	|- D = (om \ {(/)})
bnj1016.14	|- B = {f | E.n e. D (f Fn n /\ ph /\ ps)}
bnj1016.15	|- C = U_y e. (f` m) pred(y, A, R)
bnj1016.16	|- G = (f u. {<.n, C>.})

Assertion

Ref	Expression
bnj1016	|- ((th /\ ch /\ et) -> (E.pta -> G e. B))

Distinct variable groups:   A,f,i,m,n,y   D,f,i,n   i,G,p   R,f,i,m,n,y   f,X,i,n   ch,p   et,p   f,p,n   ph,i   th,p

Proof of Theorem bnj1016

Step	Hyp	Ref	Expression
1		bnj258 12097	. . . 4 |- ((th /\ ch /\ ta /\ et) <-> ((th /\ ch /\ et) /\ ta))
2		bnj1016.1	. . . . 5 |- (ph <-> (f` (/)) = pred(X, A, R))
3		bnj1016.2	. . . . 5 |- (ps <-> A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
4		bnj1016.3	. . . . 5 |- (ch <-> (n e. D /\ f Fn n /\ ph /\ ps))
5		bnj1016.4	. . . . 5 |- (th <-> (R FrSe A /\ X e. A /\ y e. trCl(X, A, R) /\ z e. pred(y, A, R)))
6		bnj1016.5	. . . . 5 |- (ta <-> (m e. om /\ n = suc m /\ p = suc n))
7		bnj1016.7	. . . . 5 |- (ph' <-> [p / n]ph)
8		bnj1016.8	. . . . 5 |- (ps' <-> [p / n]ps)
9		bnj1016.9	. . . . 5 |- (ch' <-> [p / n]ch)
10		bnj1016.10	. . . . 5 |- (ph" <-> [G / f]ph')
11		bnj1016.11	. . . . 5 |- (ps" <-> [G / f]ps')
12		bnj1016.12	. . . . 5 |- (ch" <-> [G / f]ch')
13		bnj1016.13	. . . . 5 |- D = (om \ {(/)})
14		bnj1016.14	. . . . 5 |- B = {f | E.n e. D (f Fn n /\ ph /\ ps)}
15		bnj1016.15	. . . . 5 |- C = U_y e. (f` m) pred(y, A, R)
16		bnj1016.16	. . . . 5 |- G = (f u. {<.n, C>.})
17	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16	bnj998 13363	. . . 4 |- ((th /\ ch /\ ta /\ et) -> ch")
18	1, 17	bnj1359 13084	. . 3 |- ((th /\ ch /\ et) -> (ta -> ch"))
19	18	eximdv 1669	. 2 |- ((th /\ ch /\ et) -> (E.pta -> E.pch"))
20	4, 9, 12, 14, 16	bnj985 13359	. 2 |- (G e. B <-> E.pch")
21	19, 20	syl6ibr 230	1 |- ((th /\ ch /\ et) -> (E.pta -> G e. B))

Syntax hints:   -> wi 3   <-> wb 163   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  {cab 1871  A.wral 2105  E.wrex 2106   \ cdif 2590   u. cun 2591  (/)c0 2875  {csn 3044  <.cop 3046  U_ciun 3255  suc csuc 3659  omcom 3949   Fn wfn 3993  ` cfv 3998   /\ syn-bnj17 12019   predsyn-bnj14 12023   FrSe syn-bnj15 12027   trClsyn-bnj18 12029

This theorem is referenced by:  bnj1017 13377

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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