Theorem bnj916 13332

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
bnj916.1	|- (ph <-> (f` (/)) = pred(X, A, R))
bnj916.2	|- (ps <-> A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
bnj916.3	|- D = (om \ {(/)})
bnj916.4	|- B = {f | E.n e. D (f Fn n /\ ph /\ ps)}
bnj916.5	|- (ch <-> (f Fn n /\ ph /\ ps))

Assertion

Ref	Expression
bnj916	|- (y e. trCl(X, A, R) -> E.fE.nE.i(n e. D /\ ch /\ i e. n /\ y e. (f` i)))

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

Proof of Theorem bnj916

Step	Hyp	Ref	Expression
1		bnj916.1	. . . . . 6 |- (ph <-> (f` (/)) = pred(X, A, R))
2		bnj916.2	. . . . . 6 |- (ps <-> A.i e. om (suc i e. n -> (f` suc i) = U_y e. (f` i) pred(y, A, R)))
3	1, 2	bnj911 13331	. . . . 5 |- ((f Fn n /\ ph /\ ps) -> A.i(f Fn n /\ ph /\ ps))
4		ax-17 1317	. . . . 5 |- (y e. (f` i) -> A.n y e. (f` i))
5	3, 4	bnj912 12822	. . . 4 |- (E.nE.i(n e. D /\ (f Fn n /\ ph /\ ps) /\ i e. dom f /\ y e. (f` i)) <-> (E.n e. D (f Fn n /\ ph /\ ps) /\ E.i e. dom f y e. (f` i)))
6	5	exbii 1398	. . 3 |- (E.fE.nE.i(n e. D /\ (f Fn n /\ ph /\ ps) /\ i e. dom f /\ y e. (f` i)) <-> E.f(E.n e. D (f Fn n /\ ph /\ ps) /\ E.i e. dom f y e. (f` i)))
7		bnj916.5	. . . . 5 |- (ch <-> (f Fn n /\ ph /\ ps))
8	7	bnj913 12823	. . . 4 |- ((n e. D /\ ch /\ i e. dom f /\ y e. (f` i)) <-> (n e. D /\ (f Fn n /\ ph /\ ps) /\ i e. dom f /\ y e. (f` i)))
9	8	3exbii 1400	. . 3 |- (E.fE.nE.i(n e. D /\ ch /\ i e. dom f /\ y e. (f` i)) <-> E.fE.nE.i(n e. D /\ (f Fn n /\ ph /\ ps) /\ i e. dom f /\ y e. (f` i)))
10		bnj916.3	. . . . . . 7 |- D = (om \ {(/)})
11		bnj916.4	. . . . . . 7 |- B = {f | E.n e. D (f Fn n /\ ph /\ ps)}
12	1, 2, 10, 11	bnj882 13320	. . . . . 6 |- trCl(X, A, R) = U_f e. B U_i e. dom f(f` i)
13	12	eleq2i 1961	. . . . 5 |- (y e. trCl(X, A, R) <-> y e. U_f e. B U_i e. dom f(f` i))
14		eliun 3259	. . . . 5 |- (y e. U_f e. B U_i e. dom f(f` i) <-> E.f e. B y e. U_i e. dom f(f` i))
15		eliun 3259	. . . . . 6 |- (y e. U_i e. dom f(f` i) <-> E.i e. dom f y e. (f` i))
16	15	rexbii 2128	. . . . 5 |- (E.f e. B y e. U_i e. dom f(f` i) <-> E.f e. B E.i e. dom f y e. (f` i))
17	13, 14, 16	3bitri 194	. . . 4 |- (y e. trCl(X, A, R) <-> E.f e. B E.i e. dom f y e. (f` i))
18		df-rex 2110	. . . 4 |- (E.f e. B E.i e. dom f y e. (f` i) <-> E.f(f e. B /\ E.i e. dom f y e. (f` i)))
19	11	abeq2i 2001	. . . . . 6 |- (f e. B <-> E.n e. D (f Fn n /\ ph /\ ps))
20	19	anbi1i 539	. . . . 5 |- ((f e. B /\ E.i e. dom f y e. (f` i)) <-> (E.n e. D (f Fn n /\ ph /\ ps) /\ E.i e. dom f y e. (f` i)))
21	20	exbii 1398	. . . 4 |- (E.f(f e. B /\ E.i e. dom f y e. (f` i)) <-> E.f(E.n e. D (f Fn n /\ ph /\ ps) /\ E.i e. dom f y e. (f` i)))
22	17, 18, 21	3bitri 194	. . 3 |- (y e. trCl(X, A, R) <-> E.f(E.n e. D (f Fn n /\ ph /\ ps) /\ E.i e. dom f y e. (f` i)))
23	6, 9, 22	3bitr4ri 201	. 2 |- (y e. trCl(X, A, R) <-> E.fE.nE.i(n e. D /\ ch /\ i e. dom f /\ y e. (f` i)))
24		bnj643 12578	. . . . 5 |- ((n e. D /\ ch /\ i e. dom f /\ y e. (f` i)) -> ch)
25	7	bnj564 12542	. . . . . 6 |- (ch -> dom f = n)
26		eleq2 1958	. . . . . 6 |- (dom f = n -> (i e. dom f <-> i e. n))
27		bnj914 12824	. . . . . 6 |- ((i e. dom f <-> i e. n) -> ((n e. D /\ ch /\ i e. dom f /\ y e. (f` i)) <-> (n e. D /\ ch /\ i e. n /\ y e. (f` i))))
28	25, 26, 27	3syl 24	. . . . 5 |- (ch -> ((n e. D /\ ch /\ i e. dom f /\ y e. (f` i)) <-> (n e. D /\ ch /\ i e. n /\ y e. (f` i))))
29	24, 28	bnj915 12825	. . . 4 |- ((n e. D /\ ch /\ i e. dom f /\ y e. (f` i)) -> (n e. D /\ ch /\ i e. n /\ y e. (f` i)))
30	29	2eximi 1388	. . 3 |- (E.nE.i(n e. D /\ ch /\ i e. dom f /\ y e. (f` i)) -> E.nE.i(n e. D /\ ch /\ i e. n /\ y e. (f` i)))
31	30	eximi 1387	. 2 |- (E.fE.nE.i(n e. D /\ ch /\ i e. dom f /\ y e. (f` i)) -> E.fE.nE.i(n e. D /\ ch /\ i e. n /\ y e. (f` i)))
32	23, 31	sylbi 216	1 |- (y e. trCl(X, A, R) -> E.fE.nE.i(n e. D /\ ch /\ i e. n /\ y e. (f` i)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  E.wrex 2106   \ cdif 2590  (/)c0 2875  {csn 3044  U_ciun 3255  suc csuc 3659  omcom 3949  dom cdm 3986   Fn wfn 3993  ` cfv 3998   /\ syn-bnj17 12019   predsyn-bnj14 12023   trClsyn-bnj18 12029

This theorem is referenced by:  bnj917 13333

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-iun 3257  df-fn 4009  df-bnj17 12020  df-bnj18 12030

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