Theorem cnsscnp 7772

Description: The set of continuous functions is a subset of the set of continuous functions at a point. (Contributed by Raph Levien, 21-Oct-2006.)

Hypotheses

Ref	Expression
cnsscnp.1	|- X = U.J
cnsscnp.2	|- Y = U.K

Assertion

Ref	Expression
cnsscnp	|- ((J e. Top /\ K e. Top /\ P e. X) -> (J Cn K) (_ ((J CnP K)` P))

Proof of Theorem cnsscnp

Step	Hyp	Ref	Expression
1		fvimacnv 3805	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((Fun f /\ P e. dom f) -> ((f` P) e. y <-> P e. (`'f"y)))
2		eleq2 1535	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (x = (`'f"y) -> (P e. x <-> P e. (`'f"y)))
3	2	biimprcd 156	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (P e. (`'f"y) -> (x = (`'f"y) -> P e. x))
4	1, 3	syl6bi 214	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((Fun f /\ P e. dom f) -> ((f` P) e. y -> (x = (`'f"y) -> P e. x)))
5	4	ex 373	. . . . . . . . . . . . . . . . . . . . . 22 |- (Fun f -> (P e. dom f -> ((f` P) e. y -> (x = (`'f"y) -> P e. x))))
6	5	com4r 41	. . . . . . . . . . . . . . . . . . . . 21 |- (x = (`'f"y) -> (Fun f -> (P e. dom f -> ((f` P) e. y -> P e. x))))
7	6	com3r 35	. . . . . . . . . . . . . . . . . . . 20 |- (P e. dom f -> (x = (`'f"y) -> (Fun f -> ((f` P) e. y -> P e. x))))
8	7	imp32 363	. . . . . . . . . . . . . . . . . . 19 |- ((P e. dom f /\ (x = (`'f"y) /\ Fun f)) -> ((f` P) e. y -> P e. x))
9		inss1 2230	. . . . . . . . . . . . . . . . . . . . 21 |- (y i^i ran f) (_ y
10		imaeq2 3402	. . . . . . . . . . . . . . . . . . . . . . 23 |- (x = (`'f"y) -> (f"x) = (f"(`'f"y)))
11		funimacnv 3571	. . . . . . . . . . . . . . . . . . . . . . 23 |- (Fun f -> (f"(`'f"y)) = (y i^i ran f))
12	10, 11	sylan9eq 1527	. . . . . . . . . . . . . . . . . . . . . 22 |- ((x = (`'f"y) /\ Fun f) -> (f"x) = (y i^i ran f))
13	12	sseq1d 2088	. . . . . . . . . . . . . . . . . . . . 21 |- ((x = (`'f"y) /\ Fun f) -> ((f"x) (_ y <-> (y i^i ran f) (_ y))
14	9, 13	mpbiri 194	. . . . . . . . . . . . . . . . . . . 20 |- ((x = (`'f"y) /\ Fun f) -> (f"x) (_ y)
15	14	adantl 388	. . . . . . . . . . . . . . . . . . 19 |- ((P e. dom f /\ (x = (`'f"y) /\ Fun f)) -> (f"x) (_ y)
16	8, 15	jctird 602	. . . . . . . . . . . . . . . . . 18 |- ((P e. dom f /\ (x = (`'f"y) /\ Fun f)) -> ((f` P) e. y -> (P e. x /\ (f"x) (_ y)))
17	16	exp32 377	. . . . . . . . . . . . . . . . 17 |- (P e. dom f -> (x = (`'f"y) -> (Fun f -> ((f` P) e. y -> (P e. x /\ (f"x) (_ y)))))
18	17	com24 37	. . . . . . . . . . . . . . . 16 |- (P e. dom f -> ((f` P) e. y -> (Fun f -> (x = (`'f"y) -> (P e. x /\ (f"x) (_ y)))))
19	18	imp31 362	. . . . . . . . . . . . . . 15 |- (((P e. dom f /\ (f` P) e. y) /\ Fun f) -> (x = (`'f"y) -> (P e. x /\ (f"x) (_ y)))
20	19	r19.22sdv 1738	. . . . . . . . . . . . . 14 |- (((P e. dom f /\ (f` P) e. y) /\ Fun f) -> (E.x e. J x = (`'f"y) -> E.x e. J (P e. x /\ (f"x) (_ y)))
21		risset 1685	. . . . . . . . . . . . . 14 |- ((`'f"y) e. J <-> E.x e. J x = (`'f"y))
22	20, 21	syl5ib 206	. . . . . . . . . . . . 13 |- (((P e. dom f /\ (f` P) e. y) /\ Fun f) -> ((`'f"y) e. J -> E.x e. J (P e. x /\ (f"x) (_ y)))
23	22	exp31 376	. . . . . . . . . . . 12 |- (P e. dom f -> ((f` P) e. y -> (Fun f -> ((`'f"y) e. J -> E.x e. J (P e. x /\ (f"x) (_ y)))))
24	23	com4t 40	. . . . . . . . . . 11 |- (Fun f -> ((`'f"y) e. J -> (P e. dom f -> ((f` P) e. y -> E.x e. J (P e. x /\ (f"x) (_ y)))))
25	24	com3r 35	. . . . . . . . . 10 |- (P e. dom f -> (Fun f -> ((`'f"y) e. J -> ((f` P) e. y -> E.x e. J (P e. x /\ (f"x) (_ y)))))
26	25	imp 350	. . . . . . . . 9 |- ((P e. dom f /\ Fun f) -> ((`'f"y) e. J -> ((f` P) e. y -> E.x e. J (P e. x /\ (f"x) (_ y))))
27	26	r19.20sdv 1710	. . . . . . . 8 |- ((P e. dom f /\ Fun f) -> (A.y e. K (`'f"y) e. J -> A.y e. K ((f` P) e. y -> E.x e. J (P e. x /\ (f"x) (_ y))))
28		fdm 3631	. . . . . . . . . 10 |- (f:X-->Y -> dom f = X)
29	28	eleq2d 1541	. . . . . . . . 9 |- (f:X-->Y -> (P e. dom f <-> P e. X))
30	29	biimpar 417	. . . . . . . 8 |- ((f:X-->Y /\ P e. X) -> P e. dom f)
31		ffun 3629	. . . . . . . . 9 |- (f:X-->Y -> Fun f)
32	31	adantr 389	. . . . . . . 8 |- ((f:X-->Y /\ P e. X) -> Fun f)
33	27, 30, 32	sylanc 471	. . . . . . 7 |- ((f:X-->Y /\ P e. X) -> (A.y e. K (`'f"y) e. J -> A.y e. K ((f` P) e. y -> E.x e. J (P e. x /\ (f"x) (_ y))))
34	33	expcom 374	. . . . . 6 |- (P e. X -> (f:X-->Y -> (A.y e. K (`'f"y) e. J -> A.y e. K ((f` P) e. y -> E.x e. J (P e. x /\ (f"x) (_ y)))))
35	34	imdistand 445	. . . . 5 |- (P e. X -> ((f:X-->Y /\ A.y e. K (`'f"y) e. J) -> (f:X-->Y /\ A.y e. K ((f` P) e. y -> E.x e. J (P e. x /\ (f"x) (_ y)))))
36	35	adantl 388	. . . 4 |- (((J e. Top /\ K e. Top) /\ P e. X) -> ((f:X-->Y /\ A.y e. K (`'f"y) e. J) -> (f:X-->Y /\ A.y e. K ((f` P) e. y -> E.x e. J (P e. x /\ (f"x) (_ y)))))
37		cnsscnp.1	. . . . . 6 |- X = U.J
38		cnsscnp.2	. . . . . 6 |- Y = U.K
39	37, 38	iscn 7758	. . . . 5 |- ((J e. Top /\ K e. Top) -> (f e. (J Cn K) <-> (f:X-->Y /\ A.y e. K (`'f"y) e. J)))
40	39	adantr 389	. . . 4 |- (((J e. Top /\ K e. Top) /\ P e. X) -> (f e. (J Cn K) <-> (f:X-->Y /\ A.y e. K (`'f"y) e. J)))
41	37, 38	iscnp 7760	. . . . 5 |- ((J e. Top /\ K e. Top /\ P e. X) -> (f e. ((J CnP K)` P) <-> (f:X-->Y /\ A.y e. K ((f` P) e. y -> E.x e. J (P e. x /\ (f"x) (_ y)))))
42	41	3expa 833	. . . 4 |- (((J e. Top /\ K e. Top) /\ P e. X) -> (f e. ((J CnP K)` P) <-> (f:X-->Y /\ A.y e. K ((f` P) e. y -> E.x e. J (P e. x /\ (f"x) (_ y)))))
43	36, 40, 42	3imtr4d 543	. . 3 |- (((J e. Top /\ K e. Top) /\ P e. X) -> (f e. (J Cn K) -> f e. ((J CnP K)` P)))
44	43	3impa 828	. 2 |- ((J e. Top /\ K e. Top /\ P e. X) -> (f e. (J Cn K) -> f e. ((J CnP K)` P)))
45	44	ssrdv 2070	1 |- ((J e. Top /\ K e. Top /\ P e. X) -> (J Cn K) (_ ((J CnP K)` P))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646   i^i cin 2046   (_ wss 2047  U.cuni 2503  `'ccnv 3169  dom cdm 3170  ran crn 3171  "cima 3173  Fun wfun 3176  -->wf 3178  ` cfv 3182  (class class class)co 3963  Topctop 7588   Cn ccn 7752   CnP ccnp 7753

This theorem is referenced by:  cncnpi 7773  cncnp 7778

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-oprab 3966  df-map 4324  df-cn 7754  df-cnp 7755

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