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Theorem iscn 21771
Description: The predicate "the class 𝐹 is a continuous function from topology 𝐽 to topology 𝐾". Definition of continuous function in [Munkres] p. 102. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
iscn ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
Distinct variable groups:   𝑦,𝐽   𝑦,𝐾   𝑦,𝑋   𝑦,𝐹   𝑦,𝑌

Proof of Theorem iscn
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 cnfval 21769 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽})
21eleq2d 2895 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽}))
3 cnveq 5737 . . . . . . 7 (𝑓 = 𝐹𝑓 = 𝐹)
43imaeq1d 5921 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
54eleq1d 2894 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑦) ∈ 𝐽 ↔ (𝐹𝑦) ∈ 𝐽))
65ralbidv 3194 . . . 4 (𝑓 = 𝐹 → (∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽 ↔ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽))
76elrab 3677 . . 3 (𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽} ↔ (𝐹 ∈ (𝑌m 𝑋) ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽))
8 toponmax 21462 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
9 toponmax 21462 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
10 elmapg 8408 . . . . 5 ((𝑌𝐾𝑋𝐽) → (𝐹 ∈ (𝑌m 𝑋) ↔ 𝐹:𝑋𝑌))
118, 9, 10syl2anr 596 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝑌m 𝑋) ↔ 𝐹:𝑋𝑌))
1211anbi1d 629 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹 ∈ (𝑌m 𝑋) ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
137, 12syl5bb 284 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽} ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
142, 13bitrd 280 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  {crab 3139  ccnv 5547  cima 5551  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395  TopOnctopon 21446   Cn ccn 21760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-top 21430  df-topon 21447  df-cn 21763
This theorem is referenced by:  iscn2  21774  cnf2  21785  tgcn  21788  ssidcn  21791  iscncl  21805  cnntr  21811  cnss1  21812  cnss2  21813  cncnp  21816  cnrest  21821  cnrest2  21822  cndis  21827  cnindis  21828  kgencn  22092  kgencn3  22094  tx1cn  22145  tx2cn  22146  txdis1cn  22171  qtopid  22241  qtopcn  22250  qtopf1  22352  qustgplem  22656  ucncn  22821  cvmlift2lem9a  32447  rfcnpre1  41153  0cnf  42036
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