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Theorem iscn 23179
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 23177 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽})
21eleq2d 2822 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽}))
3 cnveq 5822 . . . . . . 7 (𝑓 = 𝐹𝑓 = 𝐹)
43imaeq1d 6018 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
54eleq1d 2821 . . . . 5 (𝑓 = 𝐹 → ((𝑓𝑦) ∈ 𝐽 ↔ (𝐹𝑦) ∈ 𝐽))
65ralbidv 3159 . . . 4 (𝑓 = 𝐹 → (∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽 ↔ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽))
76elrab 3646 . . 3 (𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑦𝐾 (𝑓𝑦) ∈ 𝐽} ↔ (𝐹 ∈ (𝑌m 𝑋) ∧ ∀𝑦𝐾 (𝐹𝑦) ∈ 𝐽))
8 toponmax 22870 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
9 toponmax 22870 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
10 elmapg 8776 . . . . 5 ((𝑌𝐾𝑋𝐽) → (𝐹 ∈ (𝑌m 𝑋) ↔ 𝐹:𝑋𝑌))
118, 9, 10syl2anr 597 . . . 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 206  wa 395   = wceq 1541  wcel 2113  wral 3051  {crab 3399  ccnv 5623  cima 5627  wf 6488  cfv 6492  (class class class)co 7358  m cmap 8763  TopOnctopon 22854   Cn ccn 23168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8765  df-top 22838  df-topon 22855  df-cn 23171
This theorem is referenced by:  iscn2  23182  cnf2  23193  tgcn  23196  ssidcn  23199  iscncl  23213  cnntr  23219  cnss1  23220  cnss2  23221  cncnp  23224  cnrest  23229  cnrest2  23230  cndis  23235  cnindis  23236  kgencn  23500  kgencn3  23502  tx1cn  23553  tx2cn  23554  txdis1cn  23579  qtopid  23649  qtopcn  23658  qtopf1  23760  qustgplem  24065  ucncn  24228  cvmlift2lem9a  35497  rfcnpre1  45264  0cnf  46121
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