Theorem cnoprab2 17007

Description: Continuity of an operation which is a function in only the second 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)}
cnoprab2.8	|- G = {<.<.y, x>., z>. | ((y e. B /\ x e. A) /\ z = R)}
cnoprab2.9	|- F e. (J Cn L)

Assertion

Ref	Expression
cnoprab2	|- G e. ((K X.t J) Cn L)

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

Proof of Theorem cnoprab2

Step	Hyp	Ref	Expression
1		cnoprab.3	. . . . 5 |- J e. Top
2		cnoprab.5	. . . . 5 |- L e. Top
3		cnoprab2.9	. . . . 5 |- F e. (J Cn L)
4		cnoprab.1	. . . . . 6 |- A = U.J
5		eqid 2170	. . . . . 6 |- U.L = U.L
6	4, 5	cnf 10054	. . . . 5 |- ((J e. Top /\ L e. Top /\ F e. (J Cn L)) -> F:A-->U.L)
7	1, 2, 3, 6	mp3an 1494	. . . 4 |- F:A-->U.L
8		ffn 4698	. . . 4 |- (F:A-->U.L -> F Fn A)
9	7, 8	ax-mp 7	. . 3 |- F Fn A
10		simpr 538	. . . 4 |- ((y e. B /\ u e. A) -> u e. A)
11		fo2nd 5179	. . . . . . . 8 |- 2nd:_V-onto->_V
12		fofn 4746	. . . . . . . 8 |- (2nd:_V-onto->_V -> 2nd Fn _V)
13	11, 12	ax-mp 7	. . . . . . 7 |- 2nd Fn _V
14		ssv 2896	. . . . . . 7 |- (B X. A) C_ _V
15		fnssres 4663	. . . . . . 7 |- ((2nd Fn _V /\ (B X. A) C_ _V) -> (2nd |` (B X. A)) Fn (B X. A))
16	13, 14, 15	mp2an 777	. . . . . 6 |- (2nd |` (B X. A)) Fn (B X. A)
17		fnoprv 5078	. . . . . 6 |- ((2nd |` (B X. A)) Fn (B X. A) <-> (2nd |` (B X. A)) = {<.<.y, u>., v>. | ((y e. B /\ u e. A) /\ v = (y(2nd |` (B X. A))u))})
18	16, 17	mpbi 254	. . . . 5 |- (2nd |` (B X. A)) = {<.<.y, u>., v>. | ((y e. B /\ u e. A) /\ v = (y(2nd |` (B X. A))u))}
19		oprvres 5096	. . . . . . . . 9 |- ((y e. B /\ u e. A) -> (y(2nd |` (B X. A))u) = (y2ndu))
20		df-opr 5022	. . . . . . . . . 10 |- (y2ndu) = (2nd` <.y, u>.)
21		visset 2572	. . . . . . . . . . 11 |- y e. _V
22		visset 2572	. . . . . . . . . . 11 |- u e. _V
23	21, 22	op2nd 5173	. . . . . . . . . 10 |- (2nd` <.y, u>.) = u
24	20, 23	eqtri 2190	. . . . . . . . 9 |- (y2ndu) = u
25	19, 24	syl6eq 2222	. . . . . . . 8 |- ((y e. B /\ u e. A) -> (y(2nd |` (B X. A))u) = u)
26	25	eqeq2d 2181	. . . . . . 7 |- ((y e. B /\ u e. A) -> (v = (y(2nd |` (B X. A))u) <-> v = u))
27	26	pm5.32i 893	. . . . . 6 |- (((y e. B /\ u e. A) /\ v = (y(2nd |` (B X. A))u)) <-> ((y e. B /\ u e. A) /\ v = u))
28	27	oprabbii 5057	. . . . 5 |- {<.<.y, u>., v>. | ((y e. B /\ u e. A) /\ v = (y(2nd |` (B X. A))u))} = {<.<.y, u>., v>. | ((y e. B /\ u e. A) /\ v = u)}
29	18, 28	eqtri 2190	. . . 4 |- (2nd |` (B X. A)) = {<.<.y, u>., v>. | ((y e. B /\ u e. A) /\ v = u)}
30		cnoprab2.8	. . . . 5 |- G = {<.<.y, x>., z>. | ((y e. B /\ x e. A) /\ z = R)}
31		ax-17 1634	. . . . . . 7 |- ((y e. B /\ u e. A) -> A.x(y e. B /\ u e. A))
32		ax-17 1634	. . . . . . . 8 |- (t e. z -> A.x t e. z)
33		cnoprab.7	. . . . . . . . . 10 |- F = {<.x, w>. | (x e. A /\ w = R)}
34		hbopab1 3755	. . . . . . . . . 10 |- (t e. {<.x, w>. | (x e. A /\ w = R)} -> A.x t e. {<.x, w>. | (x e. A /\ w = R)})
35	33, 34	hbxfr 2271	. . . . . . . . 9 |- (t e. F -> A.x t e. F)
36		ax-17 1634	. . . . . . . . 9 |- (t e. u -> A.x t e. u)
37	35, 36	hbfv 4810	. . . . . . . 8 |- (t e. (F` u) -> A.x t e. (F` u))
38	32, 37	hbeq 2274	. . . . . . 7 |- (z = (F` u) -> A.x z = (F` u))
39	31, 38	hban 1674	. . . . . 6 |- (((y e. B /\ u e. A) /\ z = (F` u)) -> A.x((y e. B /\ u e. A) /\ z = (F` u)))
40		ax-17 1634	. . . . . 6 |- (((y e. B /\ x e. A) /\ z = R) -> A.u((y e. B /\ x e. A) /\ z = R))
41		eleq1 2233	. . . . . . . . 9 |- (u = x -> (u e. A <-> x e. A))
42	41	anbi2d 814	. . . . . . . 8 |- (u = x -> ((y e. B /\ u e. A) <-> (y e. B /\ x e. A)))
43	42	anbi1d 815	. . . . . . 7 |- (u = x -> (((y e. B /\ u e. A) /\ z = (F` u)) <-> ((y e. B /\ x e. A) /\ z = (F` u))))
44		fveq2 4804	. . . . . . . . . . . 12 |- (u = x -> (F` u) = (F` x))
45	33	fveq1i 4805	. . . . . . . . . . . . 13 |- (F` x) = ({<.x, w>. | (x e. A /\ w = R)}` x)
46		cnoprab.6	. . . . . . . . . . . . . 14 |- (x e. A -> R e. X)
47		fvopab2 4879	. . . . . . . . . . . . . 14 |- ((x e. A /\ R e. X) -> ({<.x, w>. | (x e. A /\ w = R)}` x) = R)
48	46, 47	mpdan 769	. . . . . . . . . . . . 13 |- (x e. A -> ({<.x, w>. | (x e. A /\ w = R)}` x) = R)
49	45, 48	syl5eq 2214	. . . . . . . . . . . 12 |- (x e. A -> (F` x) = R)
50	44, 49	sylan9eq 2226	. . . . . . . . . . 11 |- ((u = x /\ x e. A) -> (F` u) = R)
51	50	eqeq2d 2181	. . . . . . . . . 10 |- ((u = x /\ x e. A) -> (z = (F` u) <-> z = R))
52	51	ex 494	. . . . . . . . 9 |- (u = x -> (x e. A -> (z = (F` u) <-> z = R)))
53	52	adantld 546	. . . . . . . 8 |- (u = x -> ((y e. B /\ x e. A) -> (z = (F` u) <-> z = R)))
54	53	pm5.32d 895	. . . . . . 7 |- (u = x -> (((y e. B /\ x e. A) /\ z = (F` u)) <-> ((y e. B /\ x e. A) /\ z = R)))
55	43, 54	bitrd 311	. . . . . 6 |- (u = x -> (((y e. B /\ u e. A) /\ z = (F` u)) <-> ((y e. B /\ x e. A) /\ z = R)))
56	39, 40, 55	cbvoprab2 16793	. . . . 5 |- {<.<.y, u>., z>. | ((y e. B /\ u e. A) /\ z = (F` u))} = {<.<.y, x>., z>. | ((y e. B /\ x e. A) /\ z = R)}
57	30, 56	eqtr4i 2193	. . . 4 |- G = {<.<.y, u>., z>. | ((y e. B /\ u e. A) /\ z = (F` u))}
58	10, 29, 57	oprabco 5227	. . 3 |- (F Fn A -> G = (F o. (2nd |` (B X. A))))
59	9, 58	ax-mp 7	. 2 |- G = (F o. (2nd |` (B X. A)))
60		cnoprab.4	. . . . 5 |- K e. Top
61		eqid 2170	. . . . . 6 |- (K X.t J) = (K X.t J)
62	61	txtop 9942	. . . . 5 |- ((K e. Top /\ J e. Top) -> (K X.t J) e. Top)
63	60, 1, 62	mp2an 777	. . . 4 |- (K X.t J) e. Top
64	63, 1, 2	3pm3.2i 1326	. . 3 |- ((K X.t J) e. Top /\ J e. Top /\ L e. Top)
65		cnoprab.2	. . . . . 6 |- B = U.K
66		eqid 2170	. . . . . 6 |- (B X. A) = (B X. A)
67	61, 65, 4, 66	tx2cn 11217	. . . . 5 |- ((K e. Top /\ J e. Top) -> (2nd |` (B X. A)) e. ((K X.t J) Cn J))
68	60, 1, 67	mp2an 777	. . . 4 |- (2nd |` (B X. A)) e. ((K X.t J) Cn J)
69	68, 3	pm3.2i 514	. . 3 |- ((2nd |` (B X. A)) e. ((K X.t J) Cn J) /\ F e. (J Cn L))
70		cnco 10061	. . 3 |- ((((K X.t J) e. Top /\ J e. Top /\ L e. Top) /\ ((2nd |` (B X. A)) e. ((K X.t J) Cn J) /\ F e. (J Cn L))) -> (F o. (2nd |` (B X. A))) e. ((K X.t J) Cn L))
71	64, 69, 70	mp2an 777	. 2 |- (F o. (2nd |` (B X. A))) e. ((K X.t J) Cn L)
72	59, 71	eqeltri 2243	1 |- G e. ((K X.t J) Cn L)

Syntax hints:   -> wi 3   <-> wb 231   /\ wa 433   /\ w3a 1130   = wceq 1615   e. wcel 1617  _Vcvv 2569   C_ wss 2859  <.cop 3272  U.cuni 3398  {copab 3597   X. cxp 4149   |` cres 4153   o. ccom 4155   Fn wfn 4158  -->wf 4159  -onto->wfo 4161  ` cfv 4163  (class class class)co 5020  {copab2 5021  2ndc2nd 5165  Topctop 9836   X.t ctx 9938   Cn ccn 10044

This theorem is referenced by:  cnoprab2c 17009  reparphtlem2 17149

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1619  ax-7 1621  ax-gen 1622  ax-8 1623  ax-9 1624  ax-10 1625  ax-11 1626  ax-12 1627  ax-13 1628  ax-14 1629  ax-17 1634  ax-4 1637  ax-5o 1639  ax-6o 1642  ax-9o 1792  ax-10o 1810  ax-16 1883  ax-11o 1893  ax-ext 2152  ax-rep 3628  ax-sep 3638  ax-nul 3645  ax-pow 3681  ax-pr 3719  ax-un 3961

This theorem depends on definitions:  df-bi 232  df-or 434  df-an 435  df-3an 1132  df-ex 1645  df-sb 1845  df-eu 2070  df-mo 2071  df-clab 2158  df-cleq 2163  df-clel 2166  df-ne 2297  df-ral 2389  df-rex 2390  df-rab 2392  df-v 2571  df-sbc 2731  df-csb 2806  df-dif 2862  df-un 2864  df-in 2866  df-ss 2868  df-nul 3115  df-pw 3261  df-sn 3274  df-pr 3275  df-op 3278  df-uni 3399  df-iun 3470  df-br 3540  df-opab 3598  df-id 3779  df-xp 4165  df-rel 4166  df-cnv 4167  df-co 4168  df-dm 4169  df-rn 4170  df-res 4171  df-ima 4172  df-fun 4173  df-fn 4174  df-f 4175  df-fo 4177  df-fv 4179  df-opr 5022  df-oprab 5023  df-1st 5166  df-2nd 5167  df-map 5587  df-top 9842  df-bases 9844  df-topgen 9845  df-tx 9939  df-cn 10046

Copyright terms: Public domain