Theorem bnj1408 13524

Description: Technical lemma of bnj1414 13525. 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
bnj1408.1	|- B = ( pred(X, A, R) u. U_y e. pred (X, A, R) trCl(y, A, R))
bnj1408.2	|- C = ( pred(X, A, R) u. U_y e. trCl (X, A, R) trCl(y, A, R))
bnj1408.3	|- (th <-> (R FrSe A /\ X e. A))
bnj1408.4	|- (ta <-> (B e. _V /\ TrFo(B, A, R) /\ pred(X, A, R) C_ B))

Assertion

Ref	Expression
bnj1408	|- ((R FrSe A /\ X e. A) -> trCl(X, A, R) = B)

Distinct variable groups:   y,A   y,R   y,X

Proof of Theorem bnj1408

Step	Hyp	Ref	Expression
1		bnj1408.3	. . . 4 |- (th <-> (R FrSe A /\ X e. A))
2	1	biimpri 169	. . 3 |- ((R FrSe A /\ X e. A) -> th)
3		bnj1408.4	. . . 4 |- (ta <-> (B e. _V /\ TrFo(B, A, R) /\ pred(X, A, R) C_ B))
4		bnj1408.1	. . . . 5 |- B = ( pred(X, A, R) u. U_y e. pred (X, A, R) trCl(y, A, R))
5	4	bnj1413 13523	. . . 4 |- ((R FrSe A /\ X e. A) -> B e. _V)
6	4	bnj1409 13519	. . . 4 |- ((R FrSe A /\ X e. A) -> TrFo(B, A, R))
7	4	bnj931 12837	. . . . 5 |- pred(X, A, R) C_ B
8	7	a1i 8	. . . 4 |- ((R FrSe A /\ X e. A) -> pred(X, A, R) C_ B)
9	3, 5, 6, 8	bnj1363 13090	. . 3 |- ((R FrSe A /\ X e. A) -> ta)
10	1, 3	bnj1124 13424	. . 3 |- ((th /\ ta) -> trCl(X, A, R) C_ B)
11	2, 9, 10	syl11anc 524	. 2 |- ((R FrSe A /\ X e. A) -> trCl(X, A, R) C_ B)
12		bnj906 13328	. . . . 5 |- ((R FrSe A /\ X e. A) -> pred(X, A, R) C_ trCl(X, A, R))
13		iunss1 3266	. . . . 5 |- ( pred(X, A, R) C_ trCl(X, A, R) -> U_y e. pred (X, A, R) trCl(y, A, R) C_ U_y e. trCl (X, A, R) trCl(y, A, R))
14		unss2 2777	. . . . 5 |- (U_y e. pred (X, A, R) trCl(y, A, R) C_ U_y e. trCl (X, A, R) trCl(y, A, R) -> ( pred(X, A, R) u. U_y e. pred (X, A, R) trCl(y, A, R)) C_ ( pred(X, A, R) u. U_y e. trCl (X, A, R) trCl(y, A, R)))
15	12, 13, 14	3syl 24	. . . 4 |- ((R FrSe A /\ X e. A) -> ( pred(X, A, R) u. U_y e. pred (X, A, R) trCl(y, A, R)) C_ ( pred(X, A, R) u. U_y e. trCl (X, A, R) trCl(y, A, R)))
16		bnj1408.2	. . . 4 |- C = ( pred(X, A, R) u. U_y e. trCl (X, A, R) trCl(y, A, R))
17	15, 4, 16	3sstr4g 2658	. . 3 |- ((R FrSe A /\ X e. A) -> B C_ C)
18	16	bnj1151 13436	. . 3 |- ((R FrSe A /\ X e. A) -> trCl(X, A, R) = C)
19	17, 18	sseqtr4d 2654	. 2 |- ((R FrSe A /\ X e. A) -> B C_ trCl(X, A, R))
20	11, 19	eqssd 2633	1 |- ((R FrSe A /\ X e. A) -> trCl(X, A, R) = B)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292   u. cun 2591   C_ wss 2593  U_ciun 3255   predsyn-bnj14 12023   FrSe syn-bnj15 12027   trClsyn-bnj18 12029   TrFosyn-bnj19 12031

This theorem is referenced by:  bnj1414 13525

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  ax-inf2 5731

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-reu 2111  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-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-1o 5177  df-bnj17 12020  df-bnj14 12024  df-bnj13 12026  df-bnj15 12028  df-bnj18 12030  df-bnj19 12032

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