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

Theorem iscn 21319
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 21317 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽})
21eleq2d 2830 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽}))
3 cnveq 5464 . . . . . . 7 (𝑓 = 𝐹𝑓 = 𝐹)
43imaeq1d 5647 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
54eleq1d 2829 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑦) ∈ 𝐽 ↔ (𝐹𝑦) ∈ 𝐽))
65ralbidv 3133 . . . 4 (𝑓 = 𝐹 → (∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽 ↔ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽))
76elrab 3519 . . 3 (𝐹 ∈ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽} ↔ (𝐹 ∈ (𝑌𝑚 𝑋) ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽))
8 toponmax 21010 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
9 toponmax 21010 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
10 elmapg 8073 . . . . 5 ((𝑌𝐾𝑋𝐽) → (𝐹 ∈ (𝑌𝑚 𝑋) ↔ 𝐹:𝑋𝑌))
118, 9, 10syl2anr 590 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝑌𝑚 𝑋) ↔ 𝐹:𝑋𝑌))
1211anbi1d 623 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹 ∈ (𝑌𝑚 𝑋) ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
137, 12syl5bb 274 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽} ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽)))
142, 13bitrd 270 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 197  wa 384   = wceq 1652  wcel 2155  wral 3055  {crab 3059  ccnv 5276  cima 5280  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  TopOnctopon 20994   Cn ccn 21308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-map 8062  df-top 20978  df-topon 20995  df-cn 21311
This theorem is referenced by:  iscn2  21322  cnf2  21333  tgcn  21336  ssidcn  21339  iscncl  21353  cnntr  21359  cnss1  21360  cnss2  21361  cncnp  21364  cnrest  21369  cnrest2  21370  cndis  21375  cnindis  21376  kgencn  21639  kgencn3  21641  tx1cn  21692  tx2cn  21693  txdis1cn  21718  qtopid  21788  qtopcn  21797  qtopf1  21899  qustgplem  22203  ucncn  22368  cvmlift2lem9a  31665  rfcnpre1  39762  0cnf  40660
  Copyright terms: Public domain W3C validator