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Mirrors > Home > MPE Home > Th. List > cndis | Structured version Visualization version GIF version |
Description: Every function is continuous when the domain is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
cndis | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝒫 𝐴 Cn 𝐽) = (𝑋 ↑m 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvimass 5949 | . . . . . . . 8 ⊢ (◡𝑓 “ 𝑥) ⊆ dom 𝑓 | |
2 | fdm 6522 | . . . . . . . . 9 ⊢ (𝑓:𝐴⟶𝑋 → dom 𝑓 = 𝐴) | |
3 | 2 | adantl 484 | . . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → dom 𝑓 = 𝐴) |
4 | 1, 3 | sseqtrid 4019 | . . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → (◡𝑓 “ 𝑥) ⊆ 𝐴) |
5 | elpw2g 5247 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ((◡𝑓 “ 𝑥) ∈ 𝒫 𝐴 ↔ (◡𝑓 “ 𝑥) ⊆ 𝐴)) | |
6 | 5 | ad2antrr 724 | . . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → ((◡𝑓 “ 𝑥) ∈ 𝒫 𝐴 ↔ (◡𝑓 “ 𝑥) ⊆ 𝐴)) |
7 | 4, 6 | mpbird 259 | . . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴) |
8 | 7 | ralrimivw 3183 | . . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) ∧ 𝑓:𝐴⟶𝑋) → ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴) |
9 | 8 | ex 415 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴⟶𝑋 → ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴)) |
10 | 9 | pm4.71d 564 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓:𝐴⟶𝑋 ↔ (𝑓:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴))) |
11 | toponmax 21534 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
12 | id 22 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) | |
13 | elmapg 8419 | . . . 4 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (𝑋 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝑋)) | |
14 | 11, 12, 13 | syl2anr 598 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝑋 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝑋)) |
15 | distopon 21605 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴)) | |
16 | iscn 21843 | . . . 4 ⊢ ((𝒫 𝐴 ∈ (TopOn‘𝐴) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ (𝑓:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴))) | |
17 | 15, 16 | sylan 582 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ (𝑓:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐽 (◡𝑓 “ 𝑥) ∈ 𝒫 𝐴))) |
18 | 10, 14, 17 | 3bitr4rd 314 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝑓 ∈ (𝒫 𝐴 Cn 𝐽) ↔ 𝑓 ∈ (𝑋 ↑m 𝐴))) |
19 | 18 | eqrdv 2819 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝒫 𝐴 Cn 𝐽) = (𝑋 ↑m 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 𝒫 cpw 4539 ◡ccnv 5554 dom cdm 5555 “ cima 5558 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 TopOnctopon 21518 Cn ccn 21832 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-top 21502 df-topon 21519 df-cn 21835 |
This theorem is referenced by: xkopt 22263 distgp 22707 efmndtmd 22709 symgtgp 22714 |
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