Theorem bnj1335 13503

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
bnj1335.1	|- B = {d | (d C_ A /\ A.x e. d pred(x, A, R) C_ d)}
bnj1335.2	|- Y = <.x, (f |` pred(x, A, R))>.
bnj1335.3	|- C = {f | E.d e. B (f Fn d /\ A.x e. d (f` x) = (G` Y))}
bnj1335.4	|- (ta <-> (f e. C /\ dom f = ({x} u. trCl(x, A, R))))
bnj1335.5	|- D = {x e. A | -. E.fta}
bnj1335.6	|- (ps <-> (R FrSe A /\ D =/= (/)))
bnj1335.7	|- (ch <-> (ps /\ x e. D /\ A.y e. D -. yRx))
bnj1335.8	|- (ta' <-> [y / x]ta)
bnj1335.9	|- H = {f | E.y e. pred (x, A, R)ta'}
bnj1335.10	|- (ch -> A.y e. pred (x, A, R)E!fta')
bnj1335.11	|- (et <-> ( pred(x, A, R) e. _V /\ A.y e. pred (x, A, R)E!fta' /\ H = {f | E.y e. pred (x, A, R)ta'}))

Assertion

Ref	Expression
bnj1335	|- (ch -> H e. _V)

Distinct variable groups:   A,f,x,y   R,f,x,y

Proof of Theorem bnj1335

Step	Hyp	Ref	Expression
1		bnj1335.11	. . 3 |- (et <-> ( pred(x, A, R) e. _V /\ A.y e. pred (x, A, R)E!fta' /\ H = {f | E.y e. pred (x, A, R)ta'}))
2		bnj1335.7	. . . . 5 |- (ch <-> (ps /\ x e. D /\ A.y e. D -. yRx))
3		bnj1335.6	. . . . . 6 |- (ps <-> (R FrSe A /\ D =/= (/)))
4		bnj1364 13502	. . . . . . 7 |- (R FrSe A -> R Se A)
5		df-bnj13 12026	. . . . . . 7 |- (R Se A <-> A.x e. A pred(x, A, R) e. _V)
6	4, 5	sylib 215	. . . . . 6 |- (R FrSe A -> A.x e. A pred(x, A, R) e. _V)
7	3, 6	bnj832 12706	. . . . 5 |- (ps -> A.x e. A pred(x, A, R) e. _V)
8	2, 7	bnj835 12709	. . . 4 |- (ch -> A.x e. A pred(x, A, R) e. _V)
9		bnj1335.5	. . . . 5 |- D = {x e. A | -. E.fta}
10	9, 2	bnj1212 12987	. . . 4 |- (ch -> x e. A)
11	8, 10	bnj1294 13040	. . 3 |- (ch -> pred(x, A, R) e. _V)
12		bnj1335.10	. . 3 |- (ch -> A.y e. pred (x, A, R)E!fta')
13		bnj1335.9	. . . 4 |- H = {f | E.y e. pred (x, A, R)ta'}
14	13	a1i 8	. . 3 |- (ch -> H = {f | E.y e. pred (x, A, R)ta'})
15	1, 11, 12, 14	bnj1363 13090	. 2 |- (ch -> et)
16	1	bnj1366 13091	. 2 |- (et -> H e. _V)
17	15, 16	syl 12	1 |- (ch -> H 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   class class class wbr 3338  dom cdm 3986   |` cres 3988   Fn wfn 3993  ` cfv 3998   predsyn-bnj14 12023   Se syn-bnj13 12025   FrSe syn-bnj15 12027   trClsyn-bnj18 12029

This theorem is referenced by:  bnj1365 13504

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

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-rab 2112  df-v 2294  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-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-bnj13 12026  df-bnj15 12028

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