Theorem bnj969 13351

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
bnj969.1	|- (ph <-> (f` (/)) = pred(X, A, R))
bnj969.2	|- (ps <-> A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
bnj969.3	|- (ch <-> (n e. D /\ f Fn n /\ ph /\ ps))
bnj969.10	|- D = (om \ {(/)})
bnj969.12	|- C = U_y e. (f` m) pred(y, A, R)
bnj969.14	|- (ta <-> (f Fn n /\ ph /\ ps))
bnj969.15	|- (si <-> (n e. D /\ p = suc n /\ m e. n))

Assertion

Ref	Expression
bnj969	|- (((R FrSe A /\ X e. A) /\ (ch /\ n = suc m /\ p = suc n)) -> C e. _V)

Distinct variable groups:   A,i,m,y   R,i,m,y   f,i,m,y   i,n,m

Proof of Theorem bnj969

Step	Hyp	Ref	Expression
1		simpl 346	. . . 4 |- (((R FrSe A /\ X e. A) /\ (ch /\ n = suc m /\ p = suc n)) -> (R FrSe A /\ X e. A))
2		bnj969.3	. . . . 5 |- (ch <-> (n e. D /\ f Fn n /\ ph /\ ps))
3		bnj969.14	. . . . 5 |- (ta <-> (f Fn n /\ ph /\ ps))
4	2, 3	bnj949 13341	. . . 4 |- (((R FrSe A /\ X e. A) /\ (ch /\ n = suc m /\ p = suc n)) -> ta)
5		bnj969.15	. . . . 5 |- (si <-> (n e. D /\ p = suc n /\ m e. n))
6	2, 5	bnj950 13342	. . . 4 |- (((R FrSe A /\ X e. A) /\ (ch /\ n = suc m /\ p = suc n)) -> si)
7	1, 4, 6	jca32 312	. . 3 |- (((R FrSe A /\ X e. A) /\ (ch /\ n = suc m /\ p = suc n)) -> ((R FrSe A /\ X e. A) /\ (ta /\ si)))
8		bnj256 12095	. . 3 |- ((R FrSe A /\ X e. A /\ ta /\ si) <-> ((R FrSe A /\ X e. A) /\ (ta /\ si)))
9	7, 8	sylibr 217	. 2 |- (((R FrSe A /\ X e. A) /\ (ch /\ n = suc m /\ p = suc n)) -> (R FrSe A /\ X e. A /\ ta /\ si))
10		bnj969.10	. . . 4 |- D = (om \ {(/)})
11		bnj969.1	. . . 4 |- (ph <-> (f` (/)) = pred(X, A, R))
12		bnj969.2	. . . 4 |- (ps <-> A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
13	10, 3, 5, 11, 12	bnj938 13337	. . 3 |- ((R FrSe A /\ X e. A /\ ta /\ si) -> U_y e. (f` m) pred(y, A, R) e. _V)
14		bnj969.12	. . 3 |- C = U_y e. (f` m) pred(y, A, R)
15	13, 14	syl5eqel 1975	. 2 |- ((R FrSe A /\ X e. A /\ ta /\ si) -> C e. _V)
16	9, 15	syl 12	1 |- (((R FrSe A /\ X e. A) /\ (ch /\ n = suc m /\ p = suc n)) -> C e. _V)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   \ cdif 2590  (/)c0 2875  {csn 3044  U_ciun 3255  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:  bnj910 13353  bnj1003 13368

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-fv 4014  df-bnj17 12020  df-bnj14 12024  df-bnj13 12026  df-bnj15 12028

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