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Mirrors > Home > MPE Home > Th. List > iscn2 | 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 Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
iscn.1 | ⊢ 𝑋 = ∪ 𝐽 |
iscn.2 | ⊢ 𝑌 = ∪ 𝐾 |
Ref | Expression |
---|---|
iscn2 | ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cn 21763 | . . 3 ⊢ Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (∪ 𝑘 ↑m ∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 (◡𝑓 “ 𝑦) ∈ 𝑗}) | |
2 | 1 | elmpocl 7376 | . 2 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)) |
3 | iscn.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 3 | toptopon 21453 | . . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
5 | iscn.2 | . . . 4 ⊢ 𝑌 = ∪ 𝐾 | |
6 | 5 | toptopon 21453 | . . 3 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌)) |
7 | iscn 21771 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) | |
8 | 4, 6, 7 | syl2anb 597 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) |
9 | 2, 8 | biadanii 818 | 1 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (◡𝐹 “ 𝑦) ∈ 𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {crab 3139 ∪ cuni 4830 ◡ccnv 5547 “ cima 5551 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 Topctop 21429 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: cntop1 21776 cntop2 21777 cnf 21782 cnima 21801 cnco 21802 ptpjcn 22147 |
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