Theorem bnj1498 13562

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
bnj1498.1	|- B = {d | (d C_ A /\ A.x e. d pred(x, A, R) C_ d)}
bnj1498.2	|- Y = <.x, (f |` pred(x, A, R))>.
bnj1498.3	|- C = {f | E.d e. B (f Fn d /\ A.x e. d (f` x) = (G` Y))}
bnj1498.4	|- F = U.C

Assertion

Ref	Expression
bnj1498	|- (R FrSe A -> dom F = A)

Distinct variable groups:   A,d,f,x   B,f   G,d,f,x   R,d,f,x

Proof of Theorem bnj1498

Step	Hyp	Ref	Expression
1		eliun 3259	. . . . . . 7 |- (z e. U_f e. C dom f <-> E.f e. C z e. dom f)
2		df-rex 2110	. . . . . . . 8 |- (E.f e. C z e. dom f <-> E.f(f e. C /\ z e. dom f))
3		ssel2 2616	. . . . . . . . . . 11 |- ((dom f C_ A /\ z e. dom f) -> z e. A)
4		bnj1498.3	. . . . . . . . . . . . . . . . . 18 |- C = {f | E.d e. B (f Fn d /\ A.x e. d (f` x) = (G` Y))}
5	4	bnj1436 13130	. . . . . . . . . . . . . . . . 17 |- (f e. C -> E.d e. B (f Fn d /\ A.x e. d (f` x) = (G` Y)))
6	5	bnj1299 13044	. . . . . . . . . . . . . . . 16 |- (f e. C -> E.d e. B f Fn d)
7		fndm 4512	. . . . . . . . . . . . . . . 16 |- (f Fn d -> dom f = d)
8	6, 7	bnj31 12400	. . . . . . . . . . . . . . 15 |- (f e. C -> E.d e. B dom f = d)
9	8	bnj1196 12972	. . . . . . . . . . . . . 14 |- (f e. C -> E.d(d e. B /\ dom f = d))
10		bnj1498.1	. . . . . . . . . . . . . . . 16 |- B = {d | (d C_ A /\ A.x e. d pred(x, A, R) C_ d)}
11	10	bnj1436 13130	. . . . . . . . . . . . . . 15 |- (d e. B -> (d C_ A /\ A.x e. d pred(x, A, R) C_ d))
12	11	simplld 348	. . . . . . . . . . . . . 14 |- (d e. B -> d C_ A)
13	9, 12	bnj1274 13031	. . . . . . . . . . . . 13 |- (f e. C -> E.d(d C_ A /\ dom f = d))
14		bnj1117 12927	. . . . . . . . . . . . . 14 |- ((dom f = d /\ d C_ A) -> dom f C_ A)
15	14	ancoms 484	. . . . . . . . . . . . 13 |- ((d C_ A /\ dom f = d) -> dom f C_ A)
16	13, 15	bnj593 12556	. . . . . . . . . . . 12 |- (f e. C -> E.ddom f C_ A)
17	16	bnj937 12840	. . . . . . . . . . 11 |- (f e. C -> dom f C_ A)
18	3, 17	sylan 497	. . . . . . . . . 10 |- ((f e. C /\ z e. dom f) -> z e. A)
19	18	eximi 1387	. . . . . . . . 9 |- (E.f(f e. C /\ z e. dom f) -> E.f z e. A)
20	19	bnj937 12840	. . . . . . . 8 |- (E.f(f e. C /\ z e. dom f) -> z e. A)
21	2, 20	sylbi 216	. . . . . . 7 |- (E.f e. C z e. dom f -> z e. A)
22	1, 21	sylbi 216	. . . . . 6 |- (z e. U_f e. C dom f -> z e. A)
23	4	bnj1317 13053	. . . . . . 7 |- (w e. C -> A.f w e. C)
24	23	bnj1400 13114	. . . . . 6 |- dom U. C = U_f e. C dom f
25	22, 24	eleq2s 1983	. . . . 5 |- (z e. dom U. C -> z e. A)
26		bnj1498.4	. . . . . 6 |- F = U.C
27	26	dmeqi 4158	. . . . 5 |- dom F = dom U. C
28	25, 27	eleq2s 1983	. . . 4 |- (z e. dom F -> z e. A)
29	28	ssriv 2621	. . 3 |- dom F C_ A
30	29	a1i 8	. 2 |- (R FrSe A -> dom F C_ A)
31		bnj1498.2	. . . . . . . 8 |- Y = <.x, (f |` pred(x, A, R))>.
32	10, 31, 4	bnj1493 13558	. . . . . . 7 |- (R FrSe A -> A.x e. A E.f e. C dom f = ({x} u. trCl(x, A, R)))
33		visset 2295	. . . . . . . . . . . 12 |- x e. _V
34	33	snid 3069	. . . . . . . . . . 11 |- x e. {x}
35	34	bnj959 12854	. . . . . . . . . 10 |- x e. ({x} u. trCl(x, A, R))
36		eleq2 1958	. . . . . . . . . 10 |- (dom f = ({x} u. trCl(x, A, R)) -> (x e. dom f <-> x e. ({x} u. trCl(x, A, R))))
37	35, 36	mpbiri 211	. . . . . . . . 9 |- (dom f = ({x} u. trCl(x, A, R)) -> x e. dom f)
38	37	reximi 2198	. . . . . . . 8 |- (E.f e. C dom f = ({x} u. trCl(x, A, R)) -> E.f e. C x e. dom f)
39	38	ralimi 2168	. . . . . . 7 |- (A.x e. A E.f e. C dom f = ({x} u. trCl(x, A, R)) -> A.x e. A E.f e. C x e. dom f)
40	32, 39	syl 12	. . . . . 6 |- (R FrSe A -> A.x e. A E.f e. C x e. dom f)
41		eliun 3259	. . . . . . 7 |- (x e. U_f e. C dom f <-> E.f e. C x e. dom f)
42	41	ralbii 2127	. . . . . 6 |- (A.x e. A x e. U_f e. C dom f <-> A.x e. A E.f e. C x e. dom f)
43	40, 42	sylibr 217	. . . . 5 |- (R FrSe A -> A.x e. A x e. U_f e. C dom f)
44	10, 31, 4	bnj1499 13561	. . . . 5 |- (A C_ U_f e. C dom f <-> A.x e. A x e. U_f e. C dom f)
45	43, 44	sylibr 217	. . . 4 |- (R FrSe A -> A C_ U_f e. C dom f)
46	45, 24	syl6ssr 2664	. . 3 |- (R FrSe A -> A C_ dom U. C)
47	46, 27	syl6ssr 2664	. 2 |- (R FrSe A -> A C_ dom F)
48	30, 47	eqssd 2633	1 |- (R FrSe A -> dom F = A)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  E.wrex 2106   u. cun 2591   C_ wss 2593  {csn 3044  <.cop 3046  U.cuni 3177  U_ciun 3255  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:  bnj60 13563

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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