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Theorem iscn2 21263
Description: The predicate "𝐹 is a continuous function from topology 𝐽 to topology 𝐾." Definition of continuous function in [Munkres] p. 102. (Contributed by Mario Carneiro, 21-Aug-2015.)
Hypotheses
Ref Expression
iscn.1 𝑋 = 𝐽
iscn.2 𝑌 = 𝐾
Assertion
Ref Expression
iscn2 (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
Distinct variable groups:   𝑦,𝐽   𝑦,𝐾   𝑦,𝑋   𝑦,𝐹   𝑦,𝑌

Proof of Theorem iscn2
Dummy variables 𝑓 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cn 21252 . . 3 Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ ( 𝑘𝑚 𝑗) ∣ ∀𝑦𝑘 (𝑓𝑦) ∈ 𝑗})
21elmpt2cl 7027 . 2 (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
3 iscn.1 . . . 4 𝑋 = 𝐽
43toptopon 20942 . . 3 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
5 iscn.2 . . . 4 𝑌 = 𝐾
65toptopon 20942 . . 3 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
7 iscn 21260 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
84, 6, 7syl2anb 585 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
92, 8biadan2 820 1 (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065   cuni 4575  ccnv 5249  cima 5253  wf 6026  cfv 6030  (class class class)co 6796  𝑚 cmap 8013  Topctop 20918  TopOnctopon 20935   Cn ccn 21249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-map 8015  df-top 20919  df-topon 20936  df-cn 21252
This theorem is referenced by:  cntop1  21265  cntop2  21266  cnf  21271  cnima  21290  cnco  21291  ptpjcn  21635
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