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Definition df-cn 12394
Description: Define a function on two topologies whose value is the set of continuous mappings from the first topology to the second. Based on definition of continuous function in [Munkres] p. 102. See iscn 12403 for the predicate form. (Contributed by NM, 17-Oct-2006.)
Assertion
Ref Expression
df-cn Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑦𝑘 (𝑓𝑦) ∈ 𝑗})
Distinct variable group:   𝑗,𝑘,𝑓,𝑦

Detailed syntax breakdown of Definition df-cn
StepHypRef Expression
1 ccn 12391 . 2 class Cn
2 vj . . 3 setvar 𝑗
3 vk . . 3 setvar 𝑘
4 ctop 12201 . . 3 class Top
5 vf . . . . . . . . 9 setvar 𝑓
65cv 1331 . . . . . . . 8 class 𝑓
76ccnv 4545 . . . . . . 7 class 𝑓
8 vy . . . . . . . 8 setvar 𝑦
98cv 1331 . . . . . . 7 class 𝑦
107, 9cima 4549 . . . . . 6 class (𝑓𝑦)
112cv 1331 . . . . . 6 class 𝑗
1210, 11wcel 1481 . . . . 5 wff (𝑓𝑦) ∈ 𝑗
133cv 1331 . . . . 5 class 𝑘
1412, 8, 13wral 2417 . . . 4 wff 𝑦𝑘 (𝑓𝑦) ∈ 𝑗
1513cuni 3743 . . . . 5 class 𝑘
1611cuni 3743 . . . . 5 class 𝑗
17 cmap 6549 . . . . 5 class 𝑚
1815, 16, 17co 5781 . . . 4 class ( 𝑘𝑚 𝑗)
1914, 5, 18crab 2421 . . 3 class {𝑓 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑦𝑘 (𝑓𝑦) ∈ 𝑗}
202, 3, 4, 4, 19cmpo 5783 . 2 class (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑦𝑘 (𝑓𝑦) ∈ 𝑗})
211, 20wceq 1332 1 wff Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑦𝑘 (𝑓𝑦) ∈ 𝑗})
Colors of variables: wff set class
This definition is referenced by:  cnfval  12400  iscn2  12406  ishmeo  12510
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