Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > cnima | Unicode version |
Description: An open subset of the codomain of a continuous function has an open preimage. (Contributed by FL, 15-Dec-2006.) |
Ref | Expression |
---|---|
cnima |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2137 | . . . . 5 | |
2 | eqid 2137 | . . . . 5 | |
3 | 1, 2 | iscn2 12358 | . . . 4 |
4 | 3 | simprbi 273 | . . 3 |
5 | 4 | simprd 113 | . 2 |
6 | imaeq2 4872 | . . . 4 | |
7 | 6 | eleq1d 2206 | . . 3 |
8 | 7 | rspccva 2783 | . 2 |
9 | 5, 8 | sylan 281 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2414 cuni 3731 ccnv 4533 cima 4537 wf 5114 (class class class)co 5767 ctop 12153 ccn 12343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-map 6537 df-top 12154 df-topon 12167 df-cn 12346 |
This theorem is referenced by: cnco 12379 cnclima 12381 cnntri 12382 cnss1 12384 cnss2 12385 cncnpi 12386 cnrest 12393 txcnmpt 12431 txdis1cn 12436 imasnopn 12457 hmeoima 12468 hmeoopn 12469 hmeoimaf1o 12472 |
Copyright terms: Public domain | W3C validator |