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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 12553 | . . . . . . 7 TopOn | |
2 | 1 | 3ad2ant1 1007 | . . . . . 6 TopOn |
3 | simp2 987 | . . . . . 6 TopOn | |
4 | uniexg 4411 | . . . . . . . 8 TopOn | |
5 | 4 | 3ad2ant1 1007 | . . . . . . 7 TopOn |
6 | mptexg 5704 | . . . . . . 7 | |
7 | 5, 6 | syl 14 | . . . . . 6 TopOn |
8 | unieq 3792 | . . . . . . . 8 | |
9 | 8 | oveq2d 5852 | . . . . . . . . 9 |
10 | rexeq 2660 | . . . . . . . . . . 11 | |
11 | 10 | imbi2d 229 | . . . . . . . . . 10 |
12 | 11 | ralbidv 2464 | . . . . . . . . 9 |
13 | 9, 12 | rabeqbidv 2716 | . . . . . . . 8 |
14 | 8, 13 | mpteq12dv 4058 | . . . . . . 7 |
15 | unieq 3792 | . . . . . . . . . 10 | |
16 | 15 | oveq1d 5851 | . . . . . . . . 9 |
17 | raleq 2659 | . . . . . . . . 9 | |
18 | 16, 17 | rabeqbidv 2716 | . . . . . . . 8 |
19 | 18 | mpteq2dv 4067 | . . . . . . 7 |
20 | df-cnp 12730 | . . . . . . 7 | |
21 | 14, 19, 20 | ovmpog 5967 | . . . . . 6 |
22 | 2, 3, 7, 21 | syl3anc 1227 | . . . . 5 TopOn |
23 | 22 | dmeqd 4800 | . . . 4 TopOn |
24 | eqid 2164 | . . . . 5 | |
25 | 24 | dmmptss 5094 | . . . 4 |
26 | 23, 25 | eqsstrdi 3189 | . . 3 TopOn |
27 | toponuni 12554 | . . . 4 TopOn | |
28 | 27 | 3ad2ant1 1007 | . . 3 TopOn |
29 | 26, 28 | sseqtrrd 3176 | . 2 TopOn |
30 | mptrel 4726 | . . . 4 | |
31 | 22 | releqd 4682 | . . . 4 TopOn |
32 | 30, 31 | mpbiri 167 | . . 3 TopOn |
33 | simp3 988 | . . 3 TopOn | |
34 | relelfvdm 5512 | . . 3 | |
35 | 32, 33, 34 | syl2anc 409 | . 2 TopOn |
36 | 29, 35 | sseldd 3138 | 1 TopOn |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 967 wceq 1342 wcel 2135 wral 2442 wrex 2443 crab 2446 cvv 2721 wss 3111 cuni 3783 cmpt 4037 cdm 4598 cima 4601 wrel 4603 cfv 5182 (class class class)co 5836 cmap 6605 ctop 12536 TopOnctopon 12549 ccnp 12727 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-topon 12550 df-cnp 12730 |
This theorem is referenced by: cnpf2 12748 cnptopco 12763 cncnp 12771 cnptoprest2 12781 metcnpi 13056 metcnpi2 13057 metcnpi3 13058 limccnpcntop 13185 limccnp2lem 13186 limccnp2cntop 13187 |
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