Theorem iscn 23373

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 23371	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})
2	1	eleq2d 2849	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽}))
3		cnveq 5861	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
4	3	imaeq1d 6063	. . . . . 6 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ 𝑦) = (◡𝐹 “ 𝑦))
5	4	eleq1d 2848	. . . . 5 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ 𝑦) ∈ 𝐽 ↔ (◡𝐹 “ 𝑦) ∈ 𝐽))
6	5	ralbidv 3188	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽 ↔ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))
7	6	elrab 3651	. . 3 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽} ↔ (𝐹 ∈ (𝑌 ↑m 𝑋) ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))
8		toponmax 23064	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾)
9		toponmax 23064	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
10		elmapg 8837	. . . . 5 ⊢ ((𝑌 ∈ 𝐾 ∧ 𝑋 ∈ 𝐽) → (𝐹 ∈ (𝑌 ↑m 𝑋) ↔ 𝐹:𝑋⟶𝑌))
11	8, 9, 10	syl2anr 608	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝑌 ↑m 𝑋) ↔ 𝐹:𝑋⟶𝑌))
12	11	anbi1d 642	. . 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 286	. 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 282	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416  ^◡ccnv 5662   “ cima 5666  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412   ↑_m cmap 8825  TopOnctopon 23048   Cn ccn 23362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8827  df-top 23032  df-topon 23049  df-cn 23365

This theorem is referenced by:  iscn2  23376  cnf2  23387  tgcn  23390  ssidcn  23393  iscncl  23407  cnntr  23413  cnss1  23414  cnss2  23415  cncnp  23418  cnrest  23423  cnrest2  23424  cndis  23429  cnindis  23430  kgencn  23694  kgencn3  23696  tx1cn  23747  tx2cn  23748  txdis1cn  23773  qtopid  23843  qtopcn  23852  qtopf1  23954  qustgplem  24259  ucncn  24422  cvmlift2lem9a  35776  rfcnpre1  45722  0cnf  46574