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Theorem iscn 23159

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 23157	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (𝑌 ↑_m 𝑋) ∣ ∀𝑦 ∈ 𝐾 (^◡𝑓 “ 𝑦) ∈ 𝐽})
2	1	eleq2d 2815	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ {𝑓 ∈ (𝑌 ↑_m 𝑋) ∣ ∀𝑦 ∈ 𝐾 (^◡𝑓 “ 𝑦) ∈ 𝐽}))
3		cnveq 5880	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ^◡𝑓 = ^◡𝐹)
4	3	imaeq1d 6067	. . . . . 6 ⊢ (𝑓 = 𝐹 → (^◡𝑓 “ 𝑦) = (^◡𝐹 “ 𝑦))
5	4	eleq1d 2814	. . . . 5 ⊢ (𝑓 = 𝐹 → ((^◡𝑓 “ 𝑦) ∈ 𝐽 ↔ (^◡𝐹 “ 𝑦) ∈ 𝐽))
6	5	ralbidv 3175	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ 𝐾 (^◡𝑓 “ 𝑦) ∈ 𝐽 ↔ ∀𝑦 ∈ 𝐾 (^◡𝐹 “ 𝑦) ∈ 𝐽))
7	6	elrab 3684	. . 3 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑌 ↑_m 𝑋) ∣ ∀𝑦 ∈ 𝐾 (^◡𝑓 “ 𝑦) ∈ 𝐽} ↔ (𝐹 ∈ (𝑌 ↑_m 𝑋) ∧ ∀𝑦 ∈ 𝐾 (^◡𝐹 “ 𝑦) ∈ 𝐽))
8		toponmax 22848	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾)
9		toponmax 22848	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
10		elmapg 8864	. . . . 5 ⊢ ((𝑌 ∈ 𝐾 ∧ 𝑋 ∈ 𝐽) → (𝐹 ∈ (𝑌 ↑_m 𝑋) ↔ 𝐹:𝑋⟶𝑌))
11	8, 9, 10	syl2anr 595	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝑌 ↑_m 𝑋) ↔ 𝐹:𝑋⟶𝑌))
12	11	anbi1d 629	. . 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 282	. 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 278	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (^◡𝐹 “ 𝑦) ∈ 𝐽)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 {crab 3430 ^◡ccnv 5681 “ cima 5685 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ↑_m cmap 8851 TopOnctopon 22832 Cn ccn 23148

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-map 8853 df-top 22816 df-topon 22833 df-cn 23151

This theorem is referenced by: iscn2 23162 cnf2 23173 tgcn 23176 ssidcn 23179 iscncl 23193 cnntr 23199 cnss1 23200 cnss2 23201 cncnp 23204 cnrest 23209 cnrest2 23210 cndis 23215 cnindis 23216 kgencn 23480 kgencn3 23482 tx1cn 23533 tx2cn 23534 txdis1cn 23559 qtopid 23629 qtopcn 23638 qtopf1 23740 qustgplem 24045 ucncn 24210 cvmlift2lem9a 34946 rfcnpre1 44412 0cnf 45294

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