Theorem txcnopab 10228

Description: A map into the product of two topological spaces is continuous if both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
txcnopab.1	|- T = (R X.t S)
txcnopab.2	|- W = U.U
txcnopab.3	|- H = {<.x, y>. | (x e. W /\ y = <.(F` x), (G` x)>.)}

Assertion

Ref	Expression
txcnopab	|- (((R e. Top /\ S e. Top /\ U e. Top) /\ (F e. (U Cn R) /\ G e. (U Cn S))) -> H e. (U Cn T))

Distinct variable groups:   x,T,y   x,R,y   x,S,y   x,U,y   x,W,y   x,F,y   x,G,y

Proof of Theorem txcnopab

Step	Hyp	Ref	Expression
1		txcnopab.2	. . . . . . . . 9 |- W = U.U
2		eqid 1884	. . . . . . . . 9 |- U.R = U.R
3	1, 2	cnf 9038	. . . . . . . 8 |- ((U e. Top /\ R e. Top /\ F e. (U Cn R)) -> F:W-->U.R)
4	3	3expia 1069	. . . . . . 7 |- ((U e. Top /\ R e. Top) -> (F e. (U Cn R) -> F:W-->U.R))
5	4	ancoms 484	. . . . . 6 |- ((R e. Top /\ U e. Top) -> (F e. (U Cn R) -> F:W-->U.R))
6	5	3adant2 895	. . . . 5 |- ((R e. Top /\ S e. Top /\ U e. Top) -> (F e. (U Cn R) -> F:W-->U.R))
7		eqid 1884	. . . . . . . . 9 |- U.S = U.S
8	1, 7	cnf 9038	. . . . . . . 8 |- ((U e. Top /\ S e. Top /\ G e. (U Cn S)) -> G:W-->U.S)
9	8	3expia 1069	. . . . . . 7 |- ((U e. Top /\ S e. Top) -> (G e. (U Cn S) -> G:W-->U.S))
10	9	ancoms 484	. . . . . 6 |- ((S e. Top /\ U e. Top) -> (G e. (U Cn S) -> G:W-->U.S))
11	10	3adant1 894	. . . . 5 |- ((R e. Top /\ S e. Top /\ U e. Top) -> (G e. (U Cn S) -> G:W-->U.S))
12	6, 11	anim12d 617	. . . 4 |- ((R e. Top /\ S e. Top /\ U e. Top) -> ((F e. (U Cn R) /\ G e. (U Cn S)) -> (F:W-->U.R /\ G:W-->U.S)))
13	12	imp 377	. . 3 |- (((R e. Top /\ S e. Top /\ U e. Top) /\ (F e. (U Cn R) /\ G e. (U Cn S))) -> (F:W-->U.R /\ G:W-->U.S))
14		fo1st 5032	. . . . . . . . . . . . 13 |- 1st:_V-onto->_V
15		fofn 4619	. . . . . . . . . . . . 13 |- (1st:_V-onto->_V -> 1st Fn _V)
16	14, 15	ax-mp 7	. . . . . . . . . . . 12 |- 1st Fn _V
17		ssv 2636	. . . . . . . . . . . 12 |- (U.R X. U.S) C_ _V
18		fnssres 4526	. . . . . . . . . . . 12 |- ((1st Fn _V /\ (U.R X. U.S) C_ _V) -> (1st |` (U.R X. U.S)) Fn (U.R X. U.S))
19	16, 17, 18	mp2an 761	. . . . . . . . . . 11 |- (1st |` (U.R X. U.S)) Fn (U.R X. U.S)
20	19	a1i 8	. . . . . . . . . 10 |- ((F:W-->U.R /\ G:W-->U.S) -> (1st |` (U.R X. U.S)) Fn (U.R X. U.S))
21		opelxpi 4040	. . . . . . . . . . . . . . 15 |- (((F` x) e. U.R /\ (G` x) e. U.S) -> <.(F` x), (G` x)>. e. (U.R X. U.S))
22		ffvelrn 4787	. . . . . . . . . . . . . . 15 |- ((F:W-->U.R /\ x e. W) -> (F` x) e. U.R)
23		ffvelrn 4787	. . . . . . . . . . . . . . 15 |- ((G:W-->U.S /\ x e. W) -> (G` x) e. U.S)
24	21, 22, 23	syl2an 503	. . . . . . . . . . . . . 14 |- (((F:W-->U.R /\ x e. W) /\ (G:W-->U.S /\ x e. W)) -> <.(F` x), (G` x)>. e. (U.R X. U.S))
25	24	anandirs 571	. . . . . . . . . . . . 13 |- (((F:W-->U.R /\ G:W-->U.S) /\ x e. W) -> <.(F` x), (G` x)>. e. (U.R X. U.S))
26	25	r19.21aiva 2176	. . . . . . . . . . . 12 |- ((F:W-->U.R /\ G:W-->U.S) -> A.x e. W <.(F` x), (G` x)>. e. (U.R X. U.S))
27		txcnopab.3	. . . . . . . . . . . . 13 |- H = {<.x, y>. | (x e. W /\ y = <.(F` x), (G` x)>.)}
28	27	fopab2 4796	. . . . . . . . . . . 12 |- (A.x e. W <.(F` x), (G` x)>. e. (U.R X. U.S) <-> H:W-->(U.R X. U.S))
29	26, 28	sylib 215	. . . . . . . . . . 11 |- ((F:W-->U.R /\ G:W-->U.S) -> H:W-->(U.R X. U.S))
30		ffn 4562	. . . . . . . . . . 11 |- (H:W-->(U.R X. U.S) -> H Fn W)
31	29, 30	syl 12	. . . . . . . . . 10 |- ((F:W-->U.R /\ G:W-->U.S) -> H Fn W)
32		frn 4569	. . . . . . . . . . 11 |- (H:W-->(U.R X. U.S) -> ran H C_ (U.R X. U.S))
33	29, 32	syl 12	. . . . . . . . . 10 |- ((F:W-->U.R /\ G:W-->U.S) -> ran H C_ (U.R X. U.S))
34		fnco 4521	. . . . . . . . . 10 |- (((1st |` (U.R X. U.S)) Fn (U.R X. U.S) /\ H Fn W /\ ran H C_ (U.R X. U.S)) -> ((1st |` (U.R X. U.S)) o. H) Fn W)
35	20, 31, 33, 34	syl111anc 1100	. . . . . . . . 9 |- ((F:W-->U.R /\ G:W-->U.S) -> ((1st |` (U.R X. U.S)) o. H) Fn W)
36		ffn 4562	. . . . . . . . . 10 |- (F:W-->U.R -> F Fn W)
37	36	adantr 425	. . . . . . . . 9 |- ((F:W-->U.R /\ G:W-->U.S) -> F Fn W)
38		eqfnfv 4766	. . . . . . . . 9 |- ((((1st |` (U.R X. U.S)) o. H) Fn W /\ F Fn W) -> (((1st |` (U.R X. U.S)) o. H) = F <-> (W = W /\ A.z e. W (((1st |` (U.R X. U.S)) o. H)` z) = (F` z))))
39	35, 37, 38	syl11anc 524	. . . . . . . 8 |- ((F:W-->U.R /\ G:W-->U.S) -> (((1st |` (U.R X. U.S)) o. H) = F <-> (W = W /\ A.z e. W (((1st |` (U.R X. U.S)) o. H)` z) = (F` z))))
40		eqidd 1885	. . . . . . . 8 |- ((F:W-->U.R /\ G:W-->U.S) -> W = W)
41		fnfun 4510	. . . . . . . . . . . . 13 |- ((1st |` (U.R X. U.S)) Fn (U.R X. U.S) -> Fun (1st |` (U.R X. U.S)))
42	19, 41	ax-mp 7	. . . . . . . . . . . 12 |- Fun (1st |` (U.R X. U.S))
43	42	a1i 8	. . . . . . . . . . 11 |- (((F:W-->U.R /\ G:W-->U.S) /\ z e. W) -> Fun (1st |` (U.R X. U.S)))
44	29	adantr 425	. . . . . . . . . . . 12 |- (((F:W-->U.R /\ G:W-->U.S) /\ z e. W) -> H:W-->(U.R X. U.S))
45		ffun 4565	. . . . . . . . . . . 12 |- (H:W-->(U.R X. U.S) -> Fun H)
46	44, 45	syl 12	. . . . . . . . . . 11 |- (((F:W-->U.R /\ G:W-->U.S) /\ z e. W) -> Fun H)
47		fdm 4567	. . . . . . . . . . . . . 14 |- (H:W-->(U.R X. U.S) -> dom H = W)
48	29, 47	syl 12	. . . . . . . . . . . . 13 |- ((F:W-->U.R /\ G:W-->U.S) -> dom H = W)
49	48	eleq2d 1964	. . . . . . . . . . . 12 |- ((F:W-->U.R /\ G:W-->U.S) -> (z e. dom H <-> z e. W))
50	49	biimpar 461	. . . . . . . . . . 11 |- (((F:W-->U.R /\ G:W-->U.S) /\ z e. W) -> z e. dom H)
51		fvco 4736	. . . . . . . . . . 11 |- ((Fun (1st |` (U.R X. U.S)) /\ Fun H /\ z e. dom H) -> (((1st |` (U.R X. U.S)) o. H)` z) = ((1st |` (U.R X. U.S))` (H` z)))
52	43, 46, 50, 51	syl111anc 1100	. . . . . . . . . 10 |- (((F:W-->U.R /\ G:W-->U.S) /\ z e. W) -> (((1st |` (U.R X. U.S)) o. H)` z) = ((1st |` (U.R X. U.S))` (H` z)))
53		fveq2 4681	. . . . . . . . . . . . . 14 |- (x = z -> (F` x) = (F` z))
54		fveq2 4681	. . . . . . . . . . . . . 14 |- (x = z -> (G` x) = (G` z))
55	53, 54	opeq12d 3166	. . . . . . . . . . . . 13 |- (x = z -> <.(F` x), (G` x)>. = <.(F` z), (G` z)>.)
56		opex 3527	. . . . . . . . . . . . 13 |- <.(F` z), (G` z)>. e. _V
57	55, 27, 56	fvopab4 4743	. . . . . . . . . . . 12 |- (z e. W -> (H` z) = <.(F` z), (G` z)>.)
58	57	adantl 424	. . . . . . . . . . 11 |- (((F:W-->U.R /\ G:W-->U.S) /\ z e. W) -> (H` z) = <.(F` z), (G` z)>.)
59	58	fveq2d 4685	. . . . . . . . . 10 |- (((F:W-->U.R /\ G:W-->U.S) /\ z e. W) -> ((1st |` (U.R X. U.S))` (H` z)) = ((1st |` (U.R X. U.S))` <.(F` z), (G` z)>.))
60		opelxpi 4040	. . . . . . . . . . . . . 14 |- (((F` z) e. U.R /\ (G` z) e. U.S) -> <.(F` z), (G` z)>. e. (U.R X. U.S))
61		ffvelrn 4787	. . . . . . . . . . . . . 14 |- ((F:W-->U.R /\ z e. W) -> (F` z) e. U.R)
62		ffvelrn 4787	. . . . . . . . . . . . . 14 |- ((G:W-->U.S /\ z e. W) -> (G` z) e. U.S)
63	60, 61, 62	syl2an 503	. . . . . . . . . . . . 13 |- (((F:W-->U.R /\ z e. W) /\ (G:W-->U.S /\ z e. W)) -> <.(F` z), (G` z)>. e. (U.R X. U.S))
64	63	anandirs 571	. . . . . . . . . . . 12 |- (((F:W-->U.R /\ G:W-->U.S) /\ z e. W) -> <.(F` z), (G` z)>. e. (U.R X. U.S))
65		fvres 4691	. . . . . . . . . . . 12 |- (<.(F` z), (G` z)>. e. (U.R X. U.S) -> ((1st |` (U.R X. U.S))` <.(F` z), (G` z)>.) = (1st` <.(F` z), (G` z)>.))
66	64, 65	syl 12	. . . . . . . . . . 11 |- (((F:W-->U.R /\ G:W-->U.S) /\ z e. W) -> ((1st |` (U.R X. U.S))` <.(F` z), (G` z)>.) = (1st` <.(F` z), (G` z)>.))
67		fvex 4689	. . . . . . . . . . . 12 |- (F` z) e. _V
68	67	op1st 5026	. . . . . . . . . . 11 |- (1st` <.(F` z), (G` z)>.) = (F` z)
69	66, 68	syl6eq 1944	. . . . . . . . . 10 |- (((F:W-->U.R /\ G:W-->U.S) /\ z e. W) -> ((1st |` (U.R X. U.S))` <.(F` z), (G` z)>.) = (F` z))
70	52, 59, 69	3eqtrd 1929	. . . . . . . . 9 |- (((F:W-->U.R /\ G:W-->U.S) /\ z e. W) -> (((1st |` (U.R X. U.S)) o. H)` z) = (F` z))
71	70	r19.21aiva 2176	. . . . . . . 8 |- ((F:W-->U.R /\ G:W-->U.S) -> A.z e. W (((1st |` (U.R X. U.S)) o. H)` z) = (F` z))
72	39, 40, 71	mpbir2and 802	. . . . . . 7 |- ((F:W-->U.R /\ G:W-->U.S) -> ((1st |` (U.R X. U.S)) o. H) = F)
73	72	eleq1d 1963	. . . . . 6 |- ((F:W-->U.R /\ G:W-->U.S) -> (((1st |` (U.R X. U.S)) o. H) e. (U Cn R) <-> F e. (U Cn R)))
74		fo2nd 5033	. . . . . . . . . . . . 13 |- 2nd:_V-onto->_V
75		fofn 4619	. . . . . . . . . . . . 13 |- (2nd:_V-onto->_V -> 2nd Fn _V)
76	74, 75	ax-mp 7	. . . . . . . . . . . 12 |- 2nd Fn _V
77		fnssres 4526	. . . . . . . . . . . 12 |- ((2nd Fn _V /\ (U.R X. U.S) C_ _V) -> (2nd |` (U.R X. U.S)) Fn (U.R X. U.S))
78	76, 17, 77	mp2an 761	. . . . . . . . . . 11 |- (2nd |` (U.R X. U.S)) Fn (U.R X. U.S)
79	78	a1i 8	. . . . . . . . . 10 |- ((F:W-->U.R /\ G:W-->U.S) -> (2nd |` (U.R X. U.S)) Fn (U.R X. U.S))
80		fnco 4521	. . . . . . . . . 10 |- (((2nd |` (U.R X. U.S)) Fn (U.R X. U.S) /\ H Fn W /\ ran H C_ (U.R X. U.S)) -> ((2nd |` (U.R X. U.S)) o. H) Fn W)
81	79, 31, 33, 80	syl111anc 1100	. . . . . . . . 9 |- ((F:W-->U.R /\ G:W-->U.S) -> ((2nd |` (U.R X. U.S)) o. H) Fn W)
82		ffn 4562	. . . . . . . . . 10 |- (G:W-->U.S -> G Fn W)
83	82	adantl 424	. . . . . . . . 9 |- ((F:W-->U.R /\ G:W-->U.S) -> G Fn W)
84		eqfnfv 4766	. . . . . . . . 9 |- ((((2nd |` (U.R X. U.S)) o. H) Fn W /\ G Fn W) -> (((2nd |` (U.R X. U.S)) o. H) = G <-> (W = W /\ A.z e. W (((2nd |` (U.R X. U.S)) o. H)` z) = (G` z))))
85	81, 83, 84	syl11anc 524	. . . . . . . 8 |- ((F:W-->U.R /\ G:W-->U.S) -> (((2nd |` (U.R X. U.S)) o. H) = G <-> (W = W /\ A.z e. W (((2nd |` (U.R X. U.S)) o. H)` z) = (G` z))))
86		fnfun 4510	. . . . . . . . . . . . 13 |- ((2nd |` (U.R X. U.S)) Fn (U.R X. U.S) -> Fun (2nd |` (U.R X. U.S)))
87	78, 86	ax-mp 7	. . . . . . . . . . . 12 |- Fun (2nd |` (U.R X. U.S))
88	87	a1i 8	. . . . . . . . . . 11 |- (((F:W-->U.R /\ G:W-->U.S) /\ z e. W) -> Fun (2nd |` (U.R X. U.S)))
89		fvco 4736	. . . . . . . . . . 11 |- ((Fun (2nd |` (U.R X. U.S)) /\ Fun H /\ z e. dom H) -> (((2nd |` (U.R X. U.S)) o. H)` z) = ((2nd |` (U.R X. U.S))` (H` z)))
90	88, 46, 50, 89	syl111anc 1100	. . . . . . . . . 10 |- (((F:W-->U.R /\ G:W-->U.S) /\ z e. W) -> (((2nd |` (U.R X. U.S)) o. H)` z) = ((2nd |` (U.R X. U.S))` (H` z)))
91	58	fveq2d 4685	. . . . . . . . . 10 |- (((F:W-->U.R /\ G:W-->U.S) /\ z e. W) -> ((2nd |` (U.R X. U.S))` (H` z)) = ((2nd |` (U.R X. U.S))` <.(F` z), (G` z)>.))
92		fvres 4691	. . . . . . . . . . . 12 |- (<.(F` z), (G` z)>. e. (U.R X. U.S) -> ((2nd |` (U.R X. U.S))` <.(F` z), (G` z)>.) = (2nd` <.(F` z), (G` z)>.))
93	64, 92	syl 12	. . . . . . . . . . 11 |- (((F:W-->U.R /\ G:W-->U.S) /\ z e. W) -> ((2nd |` (U.R X. U.S))` <.(F` z), (G` z)>.) = (2nd` <.(F` z), (G` z)>.))
94		fvex 4689	. . . . . . . . . . . 12 |- (G` z) e. _V
95	67, 94	op2nd 5027	. . . . . . . . . . 11 |- (2nd` <.(F` z), (G` z)>.) = (G` z)
96	93, 95	syl6eq 1944	. . . . . . . . . 10 |- (((F:W-->U.R /\ G:W-->U.S) /\ z e. W) -> ((2nd |` (U.R X. U.S))` <.(F` z), (G` z)>.) = (G` z))
97	90, 91, 96	3eqtrd 1929	. . . . . . . . 9 |- (((F:W-->U.R /\ G:W-->U.S) /\ z e. W) -> (((2nd |` (U.R X. U.S)) o. H)` z) = (G` z))
98	97	r19.21aiva 2176	. . . . . . . 8 |- ((F:W-->U.R /\ G:W-->U.S) -> A.z e. W (((2nd |` (U.R X. U.S)) o. H)` z) = (G` z))
99	85, 40, 98	mpbir2and 802	. . . . . . 7 |- ((F:W-->U.R /\ G:W-->U.S) -> ((2nd |` (U.R X. U.S)) o. H) = G)
100	99	eleq1d 1963	. . . . . 6 |- ((F:W-->U.R /\ G:W-->U.S) -> (((2nd |` (U.R X. U.S)) o. H) e. (U Cn S) <-> G e. (U Cn S)))
101	73, 100	anbi12d 690	. . . . 5 |- ((F:W-->U.R /\ G:W-->U.S) -> ((((1st |` (U.R X. U.S)) o. H) e. (U Cn R) /\ ((2nd |` (U.R X. U.S)) o. H) e. (U Cn S)) <-> (F e. (U Cn R) /\ G e. (U Cn S))))
102	101	biimprcd 173	. . . 4 |- ((F e. (U Cn R) /\ G e. (U Cn S)) -> ((F:W-->U.R /\ G:W-->U.S) -> (((1st |` (U.R X. U.S)) o. H) e. (U Cn R) /\ ((2nd |` (U.R X. U.S)) o. H) e. (U Cn S))))
103	102	adantl 424	. . 3 |- (((R e. Top /\ S e. Top /\ U e. Top) /\ (F e. (U Cn R) /\ G e. (U Cn S))) -> ((F:W-->U.R /\ G:W-->U.S) -> (((1st |` (U.R X. U.S)) o. H) e. (U Cn R) /\ ((2nd |` (U.R X. U.S)) o. H) e. (U Cn S))))
104	13, 103	mpd 29	. 2 |- (((R e. Top /\ S e. Top /\ U e. Top) /\ (F e. (U Cn R) /\ G e. (U Cn S))) -> (((1st |` (U.R X. U.S)) o. H) e. (U Cn R) /\ ((2nd |` (U.R X. U.S)) o. H) e. (U Cn S)))
105	13, 29	syl 12	. . 3 |- (((R e. Top /\ S e. Top /\ U e. Top) /\ (F e. (U Cn R) /\ G e. (U Cn S))) -> H:W-->(U.R X. U.S))
106		txcnopab.1	. . . 4 |- T = (R X.t S)
107		eqid 1884	. . . 4 |- (U.R X. U.S) = (U.R X. U.S)
108		eqid 1884	. . . 4 |- (1st |` (U.R X. U.S)) = (1st |` (U.R X. U.S))
109		eqid 1884	. . . 4 |- (2nd |` (U.R X. U.S)) = (2nd |` (U.R X. U.S))
110	106, 2, 7, 107, 1, 108, 109	txcn 10227	. . 3 |- (((R e. Top /\ S e. Top /\ U e. Top) /\ H:W-->(U.R X. U.S)) -> (H e. (U Cn T) <-> (((1st |` (U.R X. U.S)) o. H) e. (U Cn R) /\ ((2nd |` (U.R X. U.S)) o. H) e. (U Cn S))))
111	105, 110	syldan 516	. 2 |- (((R e. Top /\ S e. Top /\ U e. Top) /\ (F e. (U Cn R) /\ G e. (U Cn S))) -> (H e. (U Cn T) <-> (((1st |` (U.R X. U.S)) o. H) e. (U Cn R) /\ ((2nd |` (U.R X. U.S)) o. H) e. (U Cn S))))
112	104, 111	mpbird 213	1 |- (((R e. Top /\ S e. Top /\ U e. Top) /\ (F e. (U Cn R) /\ G e. (U Cn S))) -> H e. (U Cn T))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  <.cop 3046  U.cuni 3177  {copab 3395   X. cxp 3984  dom cdm 3986  ran crn 3987   |` cres 3988   o. ccom 3990  Fun wfun 3992   Fn wfn 3993  -->wf 3994  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Topctop 8857   X.t ctx 8930   Cn ccn 9028

This theorem is referenced by:  2txcn 10229  ttcn 14913  txcnoprab 15911  cnopropabco 15917

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

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-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-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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