Theorem bnj1489 13554

Description: Technical lemma of bnj60 13563. 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
bnj1489.1	|- B = {d | (d C_ A /\ A.x e. d pred(x, A, R) C_ d)}
bnj1489.2	|- Y = <.x, (f |` pred(x, A, R))>.
bnj1489.3	|- C = {f | E.d e. B (f Fn d /\ A.x e. d (f` x) = (G` Y))}
bnj1489.4	|- (ta <-> (f e. C /\ dom f = ({x} u. trCl(x, A, R))))
bnj1489.5	|- D = {x e. A | -. E.fta}
bnj1489.6	|- (ps <-> (R FrSe A /\ D =/= (/)))
bnj1489.7	|- (ch <-> (ps /\ x e. D /\ A.y e. D -. yRx))
bnj1489.8	|- (ta' <-> [y / x]ta)
bnj1489.9	|- H = {f | E.y e. pred (x, A, R)ta'}
bnj1489.10	|- P = U.H
bnj1489.11	|- Z = <.x, (P |` pred(x, A, R))>.
bnj1489.12	|- Q = (P u. {<.x, (G` Z)>.})

Assertion

Ref	Expression
bnj1489	|- (ch -> Q e. _V)

Distinct variable groups:   A,d,f,x   y,A,f,x   B,f   y,D   G,d,f   R,d,f,x   y,R   ps,y   ta,y

Proof of Theorem bnj1489

Step	Hyp	Ref	Expression
1		bnj1489.1	. . . . . 6 |- B = {d | (d C_ A /\ A.x e. d pred(x, A, R) C_ d)}
2		bnj1489.2	. . . . . 6 |- Y = <.x, (f |` pred(x, A, R))>.
3		bnj1489.3	. . . . . 6 |- C = {f | E.d e. B (f Fn d /\ A.x e. d (f` x) = (G` Y))}
4		bnj1489.4	. . . . . 6 |- (ta <-> (f e. C /\ dom f = ({x} u. trCl(x, A, R))))
5		bnj1489.5	. . . . . 6 |- D = {x e. A | -. E.fta}
6		bnj1489.6	. . . . . 6 |- (ps <-> (R FrSe A /\ D =/= (/)))
7		bnj1489.7	. . . . . 6 |- (ch <-> (ps /\ x e. D /\ A.y e. D -. yRx))
8		bnj1489.8	. . . . . 6 |- (ta' <-> [y / x]ta)
9		bnj1489.9	. . . . . 6 |- H = {f | E.y e. pred (x, A, R)ta'}
10	1, 2, 3, 4, 5, 6, 7	bnj1487 13552	. . . . . . 7 |- (ch -> A.ych)
11	6	simplbi 349	. . . . . . . . . . 11 |- (ps -> R FrSe A)
12	7, 11	bnj835 12709	. . . . . . . . . 10 |- (ch -> R FrSe A)
13	12	adantr 425	. . . . . . . . 9 |- ((ch /\ y e. pred(x, A, R)) -> R FrSe A)
14	1, 2, 3, 4, 5, 6, 7, 8	bnj1388 13514	. . . . . . . . . 10 |- (ch -> A.y e. pred (x, A, R)E.fta')
15	14	r19.21bi 2188	. . . . . . . . 9 |- ((ch /\ y e. pred(x, A, R)) -> E.fta')
16	1, 2, 3, 4, 8	bnj1373 13506	. . . . . . . . . 10 |- (ta' <-> (f e. C /\ dom f = ({y} u. trCl(y, A, R))))
17	1, 2, 3, 16	bnj1333 13500	. . . . . . . . 9 |- ((R FrSe A /\ E.fta') -> E!fta')
18	13, 15, 17	syl11anc 524	. . . . . . . 8 |- ((ch /\ y e. pred(x, A, R)) -> E!fta')
19	18	ex 402	. . . . . . 7 |- (ch -> (y e. pred(x, A, R) -> E!fta'))
20	10, 19	r19.21ai 2174	. . . . . 6 |- (ch -> A.y e. pred (x, A, R)E!fta')
21	1, 2, 3, 4, 5, 6, 7, 8, 9, 20	bnj1365 13504	. . . . 5 |- (ch -> H e. _V)
22	21	bnj1469 13146	. . . 4 |- (ch -> U.H e. _V)
23		bnj1489.10	. . . 4 |- P = U.H
24	22, 23	syl5eqel 1975	. . 3 |- (ch -> P e. _V)
25		snex 3492	. . . 4 |- {<.x, (G` Z)>.} e. _V
26	25	a1i 8	. . 3 |- (ch -> {<.x, (G` Z)>.} e. _V)
27	24, 26	bnj1149 12944	. 2 |- (ch -> (P u. {<.x, (G` Z)>.}) e. _V)
28		bnj1489.12	. 2 |- Q = (P u. {<.x, (G` Z)>.})
29	27, 28	syl5eqel 1975	1 |- (ch -> Q e. _V)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  E!weu 1771  {cab 1871   =/= wne 2017  A.wral 2105  E.wrex 2106  {crab 2108  _Vcvv 2292   u. cun 2591   C_ wss 2593  (/)c0 2875  {csn 3044  <.cop 3046  U.cuni 3177   class class class wbr 3338  dom cdm 3986   |` cres 3988   Fn wfn 3993  ` cfv 3998   predsyn-bnj14 12023   FrSe syn-bnj15 12027   trClsyn-bnj18 12029

This theorem is referenced by:  bnj1312 13557

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

Copyright terms: Public domain