Theorem cnresoprab 15915

Description: Continuity of a restricted operation abstraction.

Hypotheses

Ref	Expression
cnresoprab.1	|- A = U.J
cnresoprab.2	|- B = U.K
cnresoprab.3	|- S = U.L
cnresoprab.4	|- C C_ A
cnresoprab.5	|- D C_ B
cnresoprab.6	|- T C_ S
cnresoprab.7	|- ((x e. C /\ y e. D) -> R e. T)
cnresoprab.8	|- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = R)}
cnresoprab.9	|- G = {<.<.x, y>., z>. | ((x e. C /\ y e. D) /\ z = R)}
cnresoprab.10	|- J e. Top
cnresoprab.11	|- K e. Top
cnresoprab.12	|- L e. Top
cnresoprab.13	|- F e. ((J X.t K) Cn L)

Assertion

Ref	Expression
cnresoprab	|- G e. ((subSp` <.(C X. D), (J X.t K)>.) Cn (subSp` <.T, L>.))

Distinct variable groups:   x,A,y,z   x,B,y,z   x,C,y,z   x,D,y,z   x,T,y   z,R

Proof of Theorem cnresoprab

Step	Hyp	Ref	Expression
1		cnresoprab.4	. . . 4 |- C C_ A
2		cnresoprab.5	. . . 4 |- D C_ B
3		resoprab2 15710	. . . 4 |- ((C C_ A /\ D C_ B) -> ({<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = R)} |` (C X. D)) = {<.<.x, y>., z>. | ((x e. C /\ y e. D) /\ z = R)})
4	1, 2, 3	mp2an 761	. . 3 |- ({<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = R)} |` (C X. D)) = {<.<.x, y>., z>. | ((x e. C /\ y e. D) /\ z = R)}
5		cnresoprab.8	. . . 4 |- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = R)}
6		reseq1 4218	. . . 4 |- (F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = R)} -> (F |` (C X. D)) = ({<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = R)} |` (C X. D)))
7	5, 6	ax-mp 7	. . 3 |- (F |` (C X. D)) = ({<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = R)} |` (C X. D))
8		cnresoprab.9	. . 3 |- G = {<.<.x, y>., z>. | ((x e. C /\ y e. D) /\ z = R)}
9	4, 7, 8	3eqtr4ri 1923	. 2 |- G = (F |` (C X. D))
10		cnresoprab.10	. . . . 5 |- J e. Top
11		cnresoprab.11	. . . . 5 |- K e. Top
12		eqid 1884	. . . . . 6 |- (J X.t K) = (J X.t K)
13	12	txtop 8934	. . . . 5 |- ((J e. Top /\ K e. Top) -> (J X.t K) e. Top)
14	10, 11, 13	mp2an 761	. . . 4 |- (J X.t K) e. Top
15		cnresoprab.12	. . . 4 |- L e. Top
16	14, 15	pm3.2i 307	. . 3 |- ((J X.t K) e. Top /\ L e. Top)
17		xpss12 4089	. . . . 5 |- ((C C_ A /\ D C_ B) -> (C X. D) C_ (A X. B))
18	1, 2, 17	mp2an 761	. . . 4 |- (C X. D) C_ (A X. B)
19		cnresoprab.6	. . . 4 |- T C_ S
20	18, 19	pm3.2i 307	. . 3 |- ((C X. D) C_ (A X. B) /\ T C_ S)
21		cnresoprab.13	. . . 4 |- F e. ((J X.t K) Cn L)
22		ax-17 1317	. . . . . . . . . 10 |- ((u e. C /\ v e. D) -> A.x(u e. C /\ v e. D))
23		visset 2295	. . . . . . . . . . . 12 |- v e. _V
24		ax-17 1317	. . . . . . . . . . . . 13 |- (w e. v -> A.x w e. v)
25		visset 2295	. . . . . . . . . . . . . 14 |- u e. _V
26		ax-17 1317	. . . . . . . . . . . . . 14 |- (w e. u -> A.x w e. u)
27	25, 26	hbcsb1 2568	. . . . . . . . . . . . 13 |- (w e. [_u / x]_R -> A.x w e. [_u / x]_R)
28	24, 27	hbcsbg 2569	. . . . . . . . . . . 12 |- (v e. _V -> (w e. [_v / y]_[_u / x]_R -> A.x w e. [_v / y]_[_u / x]_R))
29	23, 28	ax-mp 7	. . . . . . . . . . 11 |- (w e. [_v / y]_[_u / x]_R -> A.x w e. [_v / y]_[_u / x]_R)
30		ax-17 1317	. . . . . . . . . . 11 |- (w e. T -> A.x w e. T)
31	29, 30	hbel 1996	. . . . . . . . . 10 |- ([_v / y]_[_u / x]_R e. T -> A.x[_v / y]_[_u / x]_R e. T)
32	22, 31	hbim 1354	. . . . . . . . 9 |- (((u e. C /\ v e. D) -> [_v / y]_[_u / x]_R e. T) -> A.x((u e. C /\ v e. D) -> [_v / y]_[_u / x]_R e. T))
33		eleq1 1957	. . . . . . . . . . 11 |- (x = u -> (x e. C <-> u e. C))
34	33	anbi1d 679	. . . . . . . . . 10 |- (x = u -> ((x e. C /\ v e. D) <-> (u e. C /\ v e. D)))
35		csbeq1a 2546	. . . . . . . . . . . . 13 |- (x = u -> R = [_u / x]_R)
36	35	csbeq2dv 2562	. . . . . . . . . . . 12 |- ((x = u /\ v e. _V) -> [_v / y]_R = [_v / y]_[_u / x]_R)
37	23, 36	mpan2 760	. . . . . . . . . . 11 |- (x = u -> [_v / y]_R = [_v / y]_[_u / x]_R)
38	37	eleq1d 1963	. . . . . . . . . 10 |- (x = u -> ([_v / y]_R e. T <-> [_v / y]_[_u / x]_R e. T))
39	34, 38	imbi12d 688	. . . . . . . . 9 |- (x = u -> (((x e. C /\ v e. D) -> [_v / y]_R e. T) <-> ((u e. C /\ v e. D) -> [_v / y]_[_u / x]_R e. T)))
40		ax-17 1317	. . . . . . . . . . 11 |- ((x e. C /\ v e. D) -> A.y(x e. C /\ v e. D))
41		ax-17 1317	. . . . . . . . . . . . 13 |- (w e. v -> A.y w e. v)
42	23, 41	hbcsb1 2568	. . . . . . . . . . . 12 |- (w e. [_v / y]_R -> A.y w e. [_v / y]_R)
43		ax-17 1317	. . . . . . . . . . . 12 |- (w e. T -> A.y w e. T)
44	42, 43	hbel 1996	. . . . . . . . . . 11 |- ([_v / y]_R e. T -> A.y[_v / y]_R e. T)
45	40, 44	hbim 1354	. . . . . . . . . 10 |- (((x e. C /\ v e. D) -> [_v / y]_R e. T) -> A.y((x e. C /\ v e. D) -> [_v / y]_R e. T))
46		eleq1 1957	. . . . . . . . . . . 12 |- (y = v -> (y e. D <-> v e. D))
47	46	anbi2d 678	. . . . . . . . . . 11 |- (y = v -> ((x e. C /\ y e. D) <-> (x e. C /\ v e. D)))
48		csbeq1a 2546	. . . . . . . . . . . 12 |- (y = v -> R = [_v / y]_R)
49	48	eleq1d 1963	. . . . . . . . . . 11 |- (y = v -> (R e. T <-> [_v / y]_R e. T))
50	47, 49	imbi12d 688	. . . . . . . . . 10 |- (y = v -> (((x e. C /\ y e. D) -> R e. T) <-> ((x e. C /\ v e. D) -> [_v / y]_R e. T)))
51		cnresoprab.7	. . . . . . . . . 10 |- ((x e. C /\ y e. D) -> R e. T)
52	45, 50, 51	chvar 1530	. . . . . . . . 9 |- ((x e. C /\ v e. D) -> [_v / y]_R e. T)
53	32, 39, 52	chvar 1530	. . . . . . . 8 |- ((u e. C /\ v e. D) -> [_v / y]_[_u / x]_R e. T)
54	23, 41	hbcsb1 2568	. . . . . . . . . . 11 |- (w e. [_v / y]_[_u / x]_R -> A.y w e. [_v / y]_[_u / x]_R)
55		csbeq1a 2546	. . . . . . . . . . 11 |- (y = v -> [_u / x]_R = [_v / y]_[_u / x]_R)
56	27, 54, 35, 55, 5	oprabval2gf 4955	. . . . . . . . . 10 |- ((u e. A /\ v e. B /\ [_v / y]_[_u / x]_R e. T) -> (uFv) = [_v / y]_[_u / x]_R)
57	56	3expia 1069	. . . . . . . . 9 |- ((u e. A /\ v e. B) -> ([_v / y]_[_u / x]_R e. T -> (uFv) = [_v / y]_[_u / x]_R))
58	1	sseli 2617	. . . . . . . . 9 |- (u e. C -> u e. A)
59	2	sseli 2617	. . . . . . . . 9 |- (v e. D -> v e. B)
60	57, 58, 59	syl2an 503	. . . . . . . 8 |- ((u e. C /\ v e. D) -> ([_v / y]_[_u / x]_R e. T -> (uFv) = [_v / y]_[_u / x]_R))
61	53, 60	mpd 29	. . . . . . 7 |- ((u e. C /\ v e. D) -> (uFv) = [_v / y]_[_u / x]_R)
62	61, 53	eqeltrd 1971	. . . . . 6 |- ((u e. C /\ v e. D) -> (uFv) e. T)
63	62	rgen2 2186	. . . . 5 |- A.u e. C A.v e. D (uFv) e. T
64		fveq2 4681	. . . . . . . 8 |- (w = <.u, v>. -> (F` w) = (F` <.u, v>.))
65		df-opr 4886	. . . . . . . 8 |- (uFv) = (F` <.u, v>.)
66	64, 65	syl6eqr 1946	. . . . . . 7 |- (w = <.u, v>. -> (F` w) = (uFv))
67	66	eleq1d 1963	. . . . . 6 |- (w = <.u, v>. -> ((F` w) e. T <-> (uFv) e. T))
68	67	ralxp 4041	. . . . 5 |- (A.w e. (C X. D)(F` w) e. T <-> A.u e. C A.v e. D (uFv) e. T)
69	63, 68	mpbir 207	. . . 4 |- A.w e. (C X. D)(F` w) e. T
70	21, 69	pm3.2i 307	. . 3 |- (F e. ((J X.t K) Cn L) /\ A.w e. (C X. D)(F` w) e. T)
71		cnresoprab.1	. . . . . . 7 |- A = U.J
72		cnresoprab.2	. . . . . . 7 |- B = U.K
73	12, 71, 72	txuni 8935	. . . . . 6 |- ((J e. Top /\ K e. Top) -> U.(J X.t K) = (A X. B))
74	10, 11, 73	mp2an 761	. . . . 5 |- U.(J X.t K) = (A X. B)
75	74	eqcomi 1888	. . . 4 |- (A X. B) = U.(J X.t K)
76		cnresoprab.3	. . . 4 |- S = U.L
77	75, 76	cnres2 15890	. . 3 |- ((((J X.t K) e. Top /\ L e. Top) /\ ((C X. D) C_ (A X. B) /\ T C_ S) /\ (F e. ((J X.t K) Cn L) /\ A.w e. (C X. D)(F` w) e. T)) -> (F |` (C X. D)) e. ((subSp` <.(C X. D), (J X.t K)>.) Cn (subSp` <.T, L>.)))
78	16, 20, 70, 77	mp3an 1191	. 2 |- (F |` (C X. D)) e. ((subSp` <.(C X. D), (J X.t K)>.) Cn (subSp` <.T, L>.))
79	9, 78	eqeltri 1967	1 |- G e. ((subSp` <.(C X. D), (J X.t K)>.) Cn (subSp` <.T, L>.))

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  [_csb 2540   C_ wss 2593  <.cop 3046  U.cuni 3177   X. cxp 3984   |` cres 3988  ` cfv 3998  (class class class)co 4884  {copab2 4885  Topctop 8857   X.t ctx 8930   Cn ccn 9028  subSpcsubsp 10242

This theorem is referenced by:  cnresoprab2 15916  pcorevlem 16086

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

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-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  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-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-map 5383  df-top 8861  df-topsp 8862  df-bases 8863  df-topgen 8864  df-tx 8931  df-cn 9030  df-subsp 10243

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