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Mirrors > Home > MPE Home > Th. List > iscn | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
iscn | ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfval 21769 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 Cn 𝐾) = {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽}) | |
2 | 1 | eleq2d 2895 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽})) |
3 | cnveq 5737 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹) | |
4 | 3 | imaeq1d 5921 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ 𝑦) = (◡𝐹 “ 𝑦)) |
5 | 4 | eleq1d 2894 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ 𝑦) ∈ 𝐽 ↔ (◡𝐹 “ 𝑦) ∈ 𝐽)) |
6 | 5 | ralbidv 3194 | . . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽 ↔ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)) |
7 | 6 | elrab 3677 | . . 3 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 (◡𝑓 “ 𝑦) ∈ 𝐽} ↔ (𝐹 ∈ (𝑌 ↑m 𝑋) ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽)) |
8 | toponmax 21462 | . . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾) | |
9 | toponmax 21462 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
10 | elmapg 8408 | . . . . 5 ⊢ ((𝑌 ∈ 𝐾 ∧ 𝑋 ∈ 𝐽) → (𝐹 ∈ (𝑌 ↑m 𝑋) ↔ 𝐹:𝑋⟶𝑌)) | |
11 | 8, 9, 10 | syl2anr 596 | . . . 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 | syl5bb 284 | . 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 280 | 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 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {crab 3139 ◡ccnv 5547 “ cima 5551 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 TopOnctopon 21446 Cn ccn 21760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-top 21430 df-topon 21447 df-cn 21763 |
This theorem is referenced by: iscn2 21774 cnf2 21785 tgcn 21788 ssidcn 21791 iscncl 21805 cnntr 21811 cnss1 21812 cnss2 21813 cncnp 21816 cnrest 21821 cnrest2 21822 cndis 21827 cnindis 21828 kgencn 22092 kgencn3 22094 tx1cn 22145 tx2cn 22146 txdis1cn 22171 qtopid 22241 qtopcn 22250 qtopf1 22352 qustgplem 22656 ucncn 22821 cvmlift2lem9a 32447 rfcnpre1 41153 0cnf 42036 |
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