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| Mirrors > Home > MPE Home > Th. List > cnf | Structured version Visualization version GIF version | ||
| Description: A continuous function is a mapping. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
| Ref | Expression |
|---|---|
| iscnp2.1 | ⊢ 𝑋 = ∪ 𝐽 |
| iscnp2.2 | ⊢ 𝑌 = ∪ 𝐾 |
| Ref | Expression |
|---|---|
| cnf | ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iscnp2.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
| 2 | iscnp2.2 | . . . 4 ⊢ 𝑌 = ∪ 𝐾 | |
| 3 | 1, 2 | iscn2 23246 | . . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))) |
| 4 | 3 | simprbi 496 | . 2 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽)) |
| 5 | 4 | simpld 494 | 1 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶𝑌) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∪ cuni 4907 ◡ccnv 5684 “ cima 5688 ⟶wf 6557 (class class class)co 7431 Topctop 22899 Cn ccn 23232 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-top 22900 df-topon 22917 df-cn 23235 |
| This theorem is referenced by: cnco 23274 cnclima 23276 cnntri 23279 cnclsi 23280 cnss1 23284 cnss2 23285 cncnpi 23286 cncnp2 23289 cnrest 23293 cnrest2 23294 cnt0 23354 cnt1 23358 cnhaus 23362 dnsconst 23386 cncmp 23400 rncmp 23404 imacmp 23405 cnconn 23430 connima 23433 conncn 23434 2ndcomap 23466 kgencn2 23565 kgencn3 23566 txcnmpt 23632 uptx 23633 txcn 23634 hauseqlcld 23654 xkohaus 23661 xkoptsub 23662 xkopjcn 23664 xkoco1cn 23665 xkoco2cn 23666 xkococnlem 23667 cnmpt11f 23672 cnmpt21f 23680 hmeocnv 23770 hmeores 23779 txhmeo 23811 cnextfres 24077 bndth 24990 evth 24991 evth2 24992 htpyco2 25011 phtpyco2 25022 reparphti 25029 reparphtiOLD 25030 copco 25051 pcopt 25055 pcopt2 25056 pcoass 25057 pcorevlem 25059 pcorev2 25061 hauseqcn 33897 pl1cn 33954 rrhf 33999 esumcocn 34081 cnmbfm 34265 cnpconn 35235 ptpconn 35238 sconnpi1 35244 txsconnlem 35245 cvxsconn 35248 cvmseu 35281 cvmopnlem 35283 cvmfolem 35284 cvmliftmolem1 35286 cvmliftmolem2 35287 cvmliftlem3 35292 cvmliftlem6 35295 cvmliftlem7 35296 cvmliftlem8 35297 cvmliftlem9 35298 cvmliftlem10 35299 cvmliftlem11 35300 cvmliftlem13 35301 cvmliftlem15 35303 cvmlift2lem3 35310 cvmlift2lem5 35312 cvmlift2lem7 35314 cvmlift2lem9 35316 cvmlift2lem10 35317 cvmliftphtlem 35322 cvmlift3lem1 35324 cvmlift3lem2 35325 cvmlift3lem4 35327 cvmlift3lem5 35328 cvmlift3lem6 35329 cvmlift3lem7 35330 cvmlift3lem8 35331 cvmlift3lem9 35332 poimirlem31 37658 poimir 37660 broucube 37661 cnres2 37770 cnresima 37771 hausgraph 43217 refsum2cnlem1 45042 itgsubsticclem 45990 stoweidlem62 46077 cnfsmf 46755 cnneiima 48814 sepfsepc 48825 |
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