Theorem iscn 14879

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:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 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.

Step	Hyp	Ref	Expression
1		cnfval 14876	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})
2	1	eleq2d 2299	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽}))
3		cnveq 4896	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
4	3	imaeq1d 5067	. . . . . 6 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ 𝑦) = (◡𝐹 “ 𝑦))
5	4	eleq1d 2298	. . . . 5 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ 𝑦) ∈ 𝐽 ↔ (◡𝐹 “ 𝑦) ∈ 𝐽))
6	5	ralbidv 2530	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽 ↔ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))
7	6	elrab 2959	. . 3 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽} ↔ (𝐹 ∈ (𝑌 ↑𝑚 𝑋) ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))
8		toponmax 14707	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾)
9		toponmax 14707	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
10		elmapg 6816	. . . . 5 ⊢ ((𝑌 ∈ 𝐾 ∧ 𝑋 ∈ 𝐽) → (𝐹 ∈ (𝑌 ↑𝑚 𝑋) ↔ 𝐹:𝑋⟶𝑌))
11	8, 9, 10	syl2anr 290	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝑌 ↑𝑚 𝑋) ↔ 𝐹:𝑋⟶𝑌))
12	11	anbi1d 465	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹 ∈ (𝑌 ↑𝑚 𝑋) ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))
13	7, 12	bitrid 192	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽} ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))
14	2, 13	bitrd 188	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∀wral 2508  {crab 2512  ^◡ccnv 4718   “ cima 4722  ⟶wf 5314  ‘cfv 5318  (class class class)co 6007   ↑_𝑚 cmap 6803  TopOnctopon 14692   Cn ccn 14867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-map 6805  df-top 14680  df-topon 14693  df-cn 14870

This theorem is referenced by:  iscn2  14882  cnf2  14887  tgcn  14890  ssidcn  14892  cnntr  14907  cnss1  14908  cnss2  14909  cncnp  14912  cnrest  14917  cnrest2  14918  cndis  14923  tx1cn  14951  tx2cn  14952  txdis1cn  14960