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Mirrors > Home > ILE Home > Th. List > cnprcl2k | Unicode version |
Description: Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.) |
Ref | Expression |
---|---|
cnprcl2k | TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 12181 | . . . . . . 7 TopOn | |
2 | 1 | 3ad2ant1 1002 | . . . . . 6 TopOn |
3 | simp2 982 | . . . . . 6 TopOn | |
4 | uniexg 4361 | . . . . . . . 8 TopOn | |
5 | 4 | 3ad2ant1 1002 | . . . . . . 7 TopOn |
6 | mptexg 5645 | . . . . . . 7 | |
7 | 5, 6 | syl 14 | . . . . . 6 TopOn |
8 | unieq 3745 | . . . . . . . 8 | |
9 | 8 | oveq2d 5790 | . . . . . . . . 9 |
10 | rexeq 2627 | . . . . . . . . . . 11 | |
11 | 10 | imbi2d 229 | . . . . . . . . . 10 |
12 | 11 | ralbidv 2437 | . . . . . . . . 9 |
13 | 9, 12 | rabeqbidv 2681 | . . . . . . . 8 |
14 | 8, 13 | mpteq12dv 4010 | . . . . . . 7 |
15 | unieq 3745 | . . . . . . . . . 10 | |
16 | 15 | oveq1d 5789 | . . . . . . . . 9 |
17 | raleq 2626 | . . . . . . . . 9 | |
18 | 16, 17 | rabeqbidv 2681 | . . . . . . . 8 |
19 | 18 | mpteq2dv 4019 | . . . . . . 7 |
20 | df-cnp 12358 | . . . . . . 7 | |
21 | 14, 19, 20 | ovmpog 5905 | . . . . . 6 |
22 | 2, 3, 7, 21 | syl3anc 1216 | . . . . 5 TopOn |
23 | 22 | dmeqd 4741 | . . . 4 TopOn |
24 | eqid 2139 | . . . . 5 | |
25 | 24 | dmmptss 5035 | . . . 4 |
26 | 23, 25 | eqsstrdi 3149 | . . 3 TopOn |
27 | toponuni 12182 | . . . 4 TopOn | |
28 | 27 | 3ad2ant1 1002 | . . 3 TopOn |
29 | 26, 28 | sseqtrrd 3136 | . 2 TopOn |
30 | mptrel 4667 | . . . 4 | |
31 | 22 | releqd 4623 | . . . 4 TopOn |
32 | 30, 31 | mpbiri 167 | . . 3 TopOn |
33 | simp3 983 | . . 3 TopOn | |
34 | relelfvdm 5453 | . . 3 | |
35 | 32, 33, 34 | syl2anc 408 | . 2 TopOn |
36 | 29, 35 | sseldd 3098 | 1 TopOn |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 wral 2416 wrex 2417 crab 2420 cvv 2686 wss 3071 cuni 3736 cmpt 3989 cdm 4539 cima 4542 wrel 4544 cfv 5123 (class class class)co 5774 cmap 6542 ctop 12164 TopOnctopon 12177 ccnp 12355 |
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 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-topon 12178 df-cnp 12358 |
This theorem is referenced by: cnpf2 12376 cnptopco 12391 cncnp 12399 cnptoprest2 12409 metcnpi 12684 metcnpi2 12685 metcnpi3 12686 limccnpcntop 12813 limccnp2lem 12814 limccnp2cntop 12815 |
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