Theorem cnoprab1 15921

Description: Continuity of an operation which is a function in only the first variable.

Hypotheses

Ref	Expression
cnoprab.1	|- A = U.J
cnoprab.2	|- B = U.K
cnoprab.3	|- J e. Top
cnoprab.4	|- K e. Top
cnoprab.5	|- L e. Top
cnoprab.6	|- (x e. A -> R e. X)
cnoprab.7	|- F = {<.x, w>. | (x e. A /\ w = R)}
cnoprab1.8	|- G = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = R)}
cnoprab1.9	|- F e. (J Cn L)

Assertion

Ref	Expression
cnoprab1	|- G e. ((J X.t K) Cn L)

Distinct variable groups:   w,A,x,y,z   x,B,y,z   y,F,z   w,R

Proof of Theorem cnoprab1

Step	Hyp	Ref	Expression
1		cnoprab.3	. . . . 5 |- J e. Top
2		cnoprab.5	. . . . 5 |- L e. Top
3		cnoprab1.9	. . . . 5 |- F e. (J Cn L)
4		cnoprab.1	. . . . . 6 |- A = U.J
5		eqid 1884	. . . . . 6 |- U.L = U.L
6	4, 5	cnf 9038	. . . . 5 |- ((J e. Top /\ L e. Top /\ F e. (J Cn L)) -> F:A-->U.L)
7	1, 2, 3, 6	mp3an 1191	. . . 4 |- F:A-->U.L
8		ffn 4562	. . . 4 |- (F:A-->U.L -> F Fn A)
9	7, 8	ax-mp 7	. . 3 |- F Fn A
10		simpl 346	. . . 4 |- ((u e. A /\ y e. B) -> u e. A)
11		fo1st 5032	. . . . . . . 8 |- 1st:_V-onto->_V
12		fofn 4619	. . . . . . . 8 |- (1st:_V-onto->_V -> 1st Fn _V)
13	11, 12	ax-mp 7	. . . . . . 7 |- 1st Fn _V
14		ssv 2636	. . . . . . 7 |- (A X. B) C_ _V
15		fnssres 4526	. . . . . . 7 |- ((1st Fn _V /\ (A X. B) C_ _V) -> (1st |` (A X. B)) Fn (A X. B))
16	13, 14, 15	mp2an 761	. . . . . 6 |- (1st |` (A X. B)) Fn (A X. B)
17		fnoprv 4946	. . . . . 6 |- ((1st |` (A X. B)) Fn (A X. B) <-> (1st |` (A X. B)) = {<.<.u, y>., v>. | ((u e. A /\ y e. B) /\ v = (u(1st |` (A X. B))y))})
18	16, 17	mpbi 206	. . . . 5 |- (1st |` (A X. B)) = {<.<.u, y>., v>. | ((u e. A /\ y e. B) /\ v = (u(1st |` (A X. B))y))}
19		oprvres 4963	. . . . . . . . 9 |- ((u e. A /\ y e. B) -> (u(1st |` (A X. B))y) = (u1sty))
20		df-opr 4886	. . . . . . . . . 10 |- (u1sty) = (1st` <.u, y>.)
21		visset 2295	. . . . . . . . . . 11 |- u e. _V
22	21	op1st 5026	. . . . . . . . . 10 |- (1st` <.u, y>.) = u
23	20, 22	eqtri 1908	. . . . . . . . 9 |- (u1sty) = u
24	19, 23	syl6eq 1944	. . . . . . . 8 |- ((u e. A /\ y e. B) -> (u(1st |` (A X. B))y) = u)
25	24	eqeq2d 1895	. . . . . . 7 |- ((u e. A /\ y e. B) -> (v = (u(1st |` (A X. B))y) <-> v = u))
26	25	pm5.32i 707	. . . . . 6 |- (((u e. A /\ y e. B) /\ v = (u(1st |` (A X. B))y)) <-> ((u e. A /\ y e. B) /\ v = u))
27	26	oprabbii 4923	. . . . 5 |- {<.<.u, y>., v>. | ((u e. A /\ y e. B) /\ v = (u(1st |` (A X. B))y))} = {<.<.u, y>., v>. | ((u e. A /\ y e. B) /\ v = u)}
28	18, 27	eqtri 1908	. . . 4 |- (1st |` (A X. B)) = {<.<.u, y>., v>. | ((u e. A /\ y e. B) /\ v = u)}
29		cnoprab1.8	. . . . 5 |- G = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = R)}
30		ax-17 1317	. . . . . . 7 |- ((u e. A /\ y e. B) -> A.x(u e. A /\ y e. B))
31		ax-17 1317	. . . . . . . 8 |- (t e. z -> A.x t e. z)
32		cnoprab.7	. . . . . . . . . 10 |- F = {<.x, w>. | (x e. A /\ w = R)}
33		hbopab1 3562	. . . . . . . . . 10 |- (t e. {<.x, w>. | (x e. A /\ w = R)} -> A.x t e. {<.x, w>. | (x e. A /\ w = R)})
34	32, 33	hbxfr 1992	. . . . . . . . 9 |- (t e. F -> A.x t e. F)
35		ax-17 1317	. . . . . . . . 9 |- (t e. u -> A.x t e. u)
36	34, 35	hbfv 4686	. . . . . . . 8 |- (t e. (F` u) -> A.x t e. (F` u))
37	31, 36	hbeq 1995	. . . . . . 7 |- (z = (F` u) -> A.x z = (F` u))
38	30, 37	hban 1356	. . . . . 6 |- (((u e. A /\ y e. B) /\ z = (F` u)) -> A.x((u e. A /\ y e. B) /\ z = (F` u)))
39		ax-17 1317	. . . . . 6 |- (((x e. A /\ y e. B) /\ z = R) -> A.u((x e. A /\ y e. B) /\ z = R))
40		eleq1 1957	. . . . . . . . 9 |- (u = x -> (u e. A <-> x e. A))
41	40	anbi1d 679	. . . . . . . 8 |- (u = x -> ((u e. A /\ y e. B) <-> (x e. A /\ y e. B)))
42	41	anbi1d 679	. . . . . . 7 |- (u = x -> (((u e. A /\ y e. B) /\ z = (F` u)) <-> ((x e. A /\ y e. B) /\ z = (F` u))))
43		fveq2 4681	. . . . . . . . . . . 12 |- (u = x -> (F` u) = (F` x))
44		cnoprab.6	. . . . . . . . . . . . . 14 |- (x e. A -> R e. X)
45		fvopab2 4754	. . . . . . . . . . . . . 14 |- ((x e. A /\ R e. X) -> ({<.x, w>. | (x e. A /\ w = R)}` x) = R)
46	44, 45	mpdan 768	. . . . . . . . . . . . 13 |- (x e. A -> ({<.x, w>. | (x e. A /\ w = R)}` x) = R)
47	32	fveq1i 4682	. . . . . . . . . . . . 13 |- (F` x) = ({<.x, w>. | (x e. A /\ w = R)}` x)
48	46, 47	syl5eq 1940	. . . . . . . . . . . 12 |- (x e. A -> (F` x) = R)
49	43, 48	sylan9eq 1948	. . . . . . . . . . 11 |- ((u = x /\ x e. A) -> (F` u) = R)
50	49	eqeq2d 1895	. . . . . . . . . 10 |- ((u = x /\ x e. A) -> (z = (F` u) <-> z = R))
51	50	ex 402	. . . . . . . . 9 |- (u = x -> (x e. A -> (z = (F` u) <-> z = R)))
52	51	adantrd 427	. . . . . . . 8 |- (u = x -> ((x e. A /\ y e. B) -> (z = (F` u) <-> z = R)))
53	52	pm5.32d 709	. . . . . . 7 |- (u = x -> (((x e. A /\ y e. B) /\ z = (F` u)) <-> ((x e. A /\ y e. B) /\ z = R)))
54	42, 53	bitrd 587	. . . . . 6 |- (u = x -> (((u e. A /\ y e. B) /\ z = (F` u)) <-> ((x e. A /\ y e. B) /\ z = R)))
55	38, 39, 54	cbvoprab1 4924	. . . . 5 |- {<.<.u, y>., z>. | ((u e. A /\ y e. B) /\ z = (F` u))} = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = R)}
56	29, 55	eqtr4i 1911	. . . 4 |- G = {<.<.u, y>., z>. | ((u e. A /\ y e. B) /\ z = (F` u))}
57	10, 28, 56	oprabco 10159	. . 3 |- (F Fn A -> G = (F o. (1st |` (A X. B))))
58	9, 57	ax-mp 7	. 2 |- G = (F o. (1st |` (A X. B)))
59		cnoprab.4	. . . . 5 |- K e. Top
60		eqid 1884	. . . . . 6 |- (J X.t K) = (J X.t K)
61	60	txtop 8934	. . . . 5 |- ((J e. Top /\ K e. Top) -> (J X.t K) e. Top)
62	1, 59, 61	mp2an 761	. . . 4 |- (J X.t K) e. Top
63	62, 1, 2	3pm3.2i 1048	. . 3 |- ((J X.t K) e. Top /\ J e. Top /\ L e. Top)
64		cnoprab.2	. . . . . 6 |- B = U.K
65		eqid 1884	. . . . . 6 |- (A X. B) = (A X. B)
66	60, 4, 64, 65	tx1cn 10223	. . . . 5 |- ((J e. Top /\ K e. Top) -> (1st |` (A X. B)) e. ((J X.t K) Cn J))
67	1, 59, 66	mp2an 761	. . . 4 |- (1st |` (A X. B)) e. ((J X.t K) Cn J)
68	67, 3	pm3.2i 307	. . 3 |- ((1st |` (A X. B)) e. ((J X.t K) Cn J) /\ F e. (J Cn L))
69		cnco 9045	. . 3 |- ((((J X.t K) e. Top /\ J e. Top /\ L e. Top) /\ ((1st |` (A X. B)) e. ((J X.t K) Cn J) /\ F e. (J Cn L))) -> (F o. (1st |` (A X. B))) e. ((J X.t K) Cn L))
70	63, 68, 69	mp2an 761	. 2 |- (F o. (1st |` (A X. B))) e. ((J X.t K) Cn L)
71	58, 70	eqeltri 1967	1 |- G e. ((J X.t K) Cn L)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292   C_ wss 2593  <.cop 3046  U.cuni 3177  {copab 3395   X. cxp 3984   |` cres 3988   o. ccom 3990   Fn wfn 3993  -->wf 3994  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  {copab2 4885  1stc1st 5018  Topctop 8857   X.t ctx 8930   Cn ccn 9028

This theorem is referenced by:  cnoprab1c 15923  reparphtlem2 16064

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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-sbc 2454  df-csb 2541  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-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-fo 4012  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-map 5383  df-top 8861  df-bases 8863  df-topgen 8864  df-tx 8931  df-cn 9030

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