Theorem bnj999 13365

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
bnj999.1	|- (ph <-> (f` (/)) = pred(X, A, R))
bnj999.2	|- (ps <-> A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
bnj999.3	|- (ch <-> (n e. D /\ f Fn n /\ ph /\ ps))
bnj999.7	|- (ph' <-> [p / n]ph)
bnj999.8	|- (ps' <-> [p / n]ps)
bnj999.9	|- (ch' <-> [p / n]ch)
bnj999.10	|- (ph" <-> [G / f]ph')
bnj999.11	|- (ps" <-> [G / f]ps')
bnj999.12	|- (ch" <-> [G / f]ch')
bnj999.15	|- C = U_y e. (f` m) pred(y, A, R)
bnj999.16	|- G = (f u. {<.n, C>.})

Assertion

Ref	Expression
bnj999	|- ((ch" /\ i e. om /\ suc i e. p /\ y e. (G` i)) -> pred(y, A, R) C_ (G` suc i))

Distinct variable groups:   f,i,n,y   A,f,n   D,f,n   i,G   R,f,n   f,X,n   f,p,i,n

Proof of Theorem bnj999

Step	Hyp	Ref	Expression
1		bnj999.3	. . . . . . 7 |- (ch <-> (n e. D /\ f Fn n /\ ph /\ ps))
2		bnj999.7	. . . . . . 7 |- (ph' <-> [p / n]ph)
3		bnj999.8	. . . . . . 7 |- (ps' <-> [p / n]ps)
4		bnj999.9	. . . . . . 7 |- (ch' <-> [p / n]ch)
5		visset 2295	. . . . . . 7 |- p e. _V
6	1, 2, 3, 4, 5	bnj919 12829	. . . . . 6 |- (ch' <-> (p e. D /\ f Fn p /\ ph' /\ ps'))
7		bnj999.10	. . . . . 6 |- (ph" <-> [G / f]ph')
8		bnj999.11	. . . . . 6 |- (ps" <-> [G / f]ps')
9		bnj999.12	. . . . . 6 |- (ch" <-> [G / f]ch')
10		bnj999.16	. . . . . . 7 |- G = (f u. {<.n, C>.})
11	10	bnj918 12826	. . . . . 6 |- G e. _V
12	6, 7, 8, 9, 11	bnj976 12861	. . . . 5 |- (ch" <-> (p e. D /\ G Fn p /\ ph" /\ ps"))
13	12	bnj1254 13018	. . . 4 |- (ch" -> ps")
14	13	anim1i 361	. . 3 |- ((ch" /\ (i e. om /\ suc i e. p /\ y e. (G` i))) -> (ps" /\ (i e. om /\ suc i e. p /\ y e. (G` i))))
15		bnj252 12091	. . 3 |- ((ch" /\ i e. om /\ suc i e. p /\ y e. (G` i)) <-> (ch" /\ (i e. om /\ suc i e. p /\ y e. (G` i))))
16		bnj252 12091	. . 3 |- ((ps" /\ i e. om /\ suc i e. p /\ y e. (G` i)) <-> (ps" /\ (i e. om /\ suc i e. p /\ y e. (G` i))))
17	14, 15, 16	3imtr4i 236	. 2 |- ((ch" /\ i e. om /\ suc i e. p /\ y e. (G` i)) -> (ps" /\ i e. om /\ suc i e. p /\ y e. (G` i)))
18		ssiun2 3295	. . . 4 |- (y e. (G` i) -> pred(y, A, R) C_ U_y e. (G` i) pred(y, A, R))
19	18	bnj708 12646	. . 3 |- ((ps" /\ i e. om /\ suc i e. p /\ y e. (G` i)) -> pred(y, A, R) C_ U_y e. (G` i) pred(y, A, R))
20		bnj633 12569	. . . . 5 |- ((ps" /\ i e. om /\ suc i e. p) -> (i e. om /\ ps"))
21		simp3 878	. . . . 5 |- ((ps" /\ i e. om /\ suc i e. p) -> suc i e. p)
22		bnj999.2	. . . . . . . 8 |- (ps <-> A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
23	22, 3, 5	bnj539 13266	. . . . . . 7 |- (ps' <-> A.i e. om (suc i e. p -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
24		bnj999.15	. . . . . . 7 |- C = U_y e. (f` m) pred(y, A, R)
25	23, 8, 24, 10	bnj965 13346	. . . . . 6 |- (ps" <-> A.i e. om (suc i e. p -> (G` suc i) = U_y e. (G` i) pred(y, A, R)))
26	25	bnj228 12517	. . . . 5 |- ((i e. om /\ ps") -> (suc i e. p -> (G` suc i) = U_y e. (G` i) pred(y, A, R)))
27	20, 21, 26	sylc 83	. . . 4 |- ((ps" /\ i e. om /\ suc i e. p) -> (G` suc i) = U_y e. (G` i) pred(y, A, R))
28	27	bnj721 12659	. . 3 |- ((ps" /\ i e. om /\ suc i e. p /\ y e. (G` i)) -> (G` suc i) = U_y e. (G` i) pred(y, A, R))
29	19, 28	sseqtr4d 2654	. 2 |- ((ps" /\ i e. om /\ suc i e. p /\ y e. (G` i)) -> pred(y, A, R) C_ (G` suc i))
30	17, 29	syl 12	1 |- ((ch" /\ i e. om /\ suc i e. p /\ y e. (G` i)) -> pred(y, A, R) C_ (G` suc i))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105   u. cun 2591   C_ wss 2593  (/)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

This theorem is referenced by:  bnj1002 13367

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-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  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-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

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