MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscn Structured version   Visualization version   GIF version

Theorem iscn 20944
Description: The predicate "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 20942 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽})
21eleq2d 2689 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽}))
3 cnveq 5261 . . . . . . 7 (𝑓 = 𝐹𝑓 = 𝐹)
43imaeq1d 5428 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
54eleq1d 2688 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑦) ∈ 𝐽 ↔ (𝐹𝑦) ∈ 𝐽))
65ralbidv 2985 . . . 4 (𝑓 = 𝐹 → (∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽 ↔ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽))
76elrab 3351 . . 3 (𝐹 ∈ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽} ↔ (𝐹 ∈ (𝑌𝑚 𝑋) ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽))
8 toponmax 20638 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
9 toponmax 20638 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
10 elmapg 7816 . . . . 5 ((𝑌𝐾𝑋𝐽) → (𝐹 ∈ (𝑌𝑚 𝑋) ↔ 𝐹:𝑋𝑌))
118, 9, 10syl2anr 495 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝑌𝑚 𝑋) ↔ 𝐹:𝑋𝑌))
1211anbi1d 740 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹 ∈ (𝑌𝑚 𝑋) ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
137, 12syl5bb 272 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽} ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
142, 13bitrd 268 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 196  wa 384   = wceq 1480  wcel 1992  wral 2912  {crab 2916  ccnv 5078  cima 5082  wf 5846  cfv 5850  (class class class)co 6605  𝑚 cmap 7803  TopOnctopon 20613   Cn ccn 20933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-top 20616  df-topon 20618  df-cn 20936
This theorem is referenced by:  iscn2  20947  cnf2  20958  tgcn  20961  ssidcn  20964  iscncl  20978  cnntr  20984  cnss1  20985  cnss2  20986  cncnp  20989  cnrest  20994  cnrest2  20995  cndis  21000  cnindis  21001  kgencn  21264  kgencn3  21266  tx1cn  21317  tx2cn  21318  txdis1cn  21343  qtopid  21413  qtopcn  21422  qtopf1  21524  qustgplem  21829  ucncn  21994  cvmlift2lem9a  30985  rfcnpre1  38647  0cnf  39380
  Copyright terms: Public domain W3C validator