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Theorem iscn 22514
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 22512 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽})
21eleq2d 2824 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽}))
3 cnveq 5826 . . . . . . 7 (𝑓 = 𝐹𝑓 = 𝐹)
43imaeq1d 6009 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
54eleq1d 2823 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑦) ∈ 𝐽 ↔ (𝐹𝑦) ∈ 𝐽))
65ralbidv 3173 . . . 4 (𝑓 = 𝐹 → (∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽 ↔ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽))
76elrab 3644 . . 3 (𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽} ↔ (𝐹 ∈ (𝑌m 𝑋) ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽))
8 toponmax 22203 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
9 toponmax 22203 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
10 elmapg 8712 . . . . 5 ((𝑌𝐾𝑋𝐽) → (𝐹 ∈ (𝑌m 𝑋) ↔ 𝐹:𝑋𝑌))
118, 9, 10syl2anr 598 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝑌m 𝑋) ↔ 𝐹:𝑋𝑌))
1211anbi1d 631 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹 ∈ (𝑌m 𝑋) ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
137, 12bitrid 283 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽} ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
142, 13bitrd 279 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 205  wa 397   = wceq 1542  wcel 2107  wral 3063  {crab 3406  ccnv 5630  cima 5634  wf 6488  cfv 6492  (class class class)co 7350  m cmap 8699  TopOnctopon 22187   Cn ccn 22503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7353  df-oprab 7354  df-mpo 7355  df-map 8701  df-top 22171  df-topon 22188  df-cn 22506
This theorem is referenced by:  iscn2  22517  cnf2  22528  tgcn  22531  ssidcn  22534  iscncl  22548  cnntr  22554  cnss1  22555  cnss2  22556  cncnp  22559  cnrest  22564  cnrest2  22565  cndis  22570  cnindis  22571  kgencn  22835  kgencn3  22837  tx1cn  22888  tx2cn  22889  txdis1cn  22914  qtopid  22984  qtopcn  22993  qtopf1  23095  qustgplem  23400  ucncn  23565  cvmlift2lem9a  33677  rfcnpre1  43025  0cnf  43909
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