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Mirrors > Home > ILE Home > Th. List > imasnopn | Unicode version |
Description: If a relation graph is open, then an image set of a singleton is also open. Corollary of Proposition 4 of [BourbakiTop1] p. I.26. (Contributed by Thierry Arnoux, 14-Jan-2018.) |
Ref | Expression |
---|---|
imasnopn.1 |
Ref | Expression |
---|---|
imasnopn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1508 | . . . 4 | |
2 | nfcv 2281 | . . . 4 | |
3 | nfrab1 2610 | . . . 4 | |
4 | txtop 12429 | . . . . . . . . . . . . 13 | |
5 | 4 | adantr 274 | . . . . . . . . . . . 12 |
6 | simprl 520 | . . . . . . . . . . . 12 | |
7 | eqid 2139 | . . . . . . . . . . . . 13 | |
8 | 7 | eltopss 12176 | . . . . . . . . . . . 12 |
9 | 5, 6, 8 | syl2anc 408 | . . . . . . . . . . 11 |
10 | imasnopn.1 | . . . . . . . . . . . . 13 | |
11 | eqid 2139 | . . . . . . . . . . . . 13 | |
12 | 10, 11 | txuni 12432 | . . . . . . . . . . . 12 |
13 | 12 | adantr 274 | . . . . . . . . . . 11 |
14 | 9, 13 | sseqtrrd 3136 | . . . . . . . . . 10 |
15 | imass1 4914 | . . . . . . . . . 10 | |
16 | 14, 15 | syl 14 | . . . . . . . . 9 |
17 | xpimasn 4987 | . . . . . . . . . 10 | |
18 | 17 | ad2antll 482 | . . . . . . . . 9 |
19 | 16, 18 | sseqtrd 3135 | . . . . . . . 8 |
20 | 19 | sseld 3096 | . . . . . . 7 |
21 | 20 | pm4.71rd 391 | . . . . . 6 |
22 | elimasng 4907 | . . . . . . . . 9 | |
23 | 22 | elvd 2691 | . . . . . . . 8 |
24 | 23 | ad2antll 482 | . . . . . . 7 |
25 | 24 | anbi2d 459 | . . . . . 6 |
26 | 21, 25 | bitrd 187 | . . . . 5 |
27 | rabid 2606 | . . . . 5 | |
28 | 26, 27 | syl6bbr 197 | . . . 4 |
29 | 1, 2, 3, 28 | eqrd 3115 | . . 3 |
30 | eqid 2139 | . . . 4 | |
31 | 30 | mptpreima 5032 | . . 3 |
32 | 29, 31 | syl6eqr 2190 | . 2 |
33 | 11 | toptopon 12185 | . . . . . 6 TopOn |
34 | 33 | biimpi 119 | . . . . 5 TopOn |
35 | 34 | ad2antlr 480 | . . . 4 TopOn |
36 | 10 | toptopon 12185 | . . . . . . 7 TopOn |
37 | 36 | biimpi 119 | . . . . . 6 TopOn |
38 | 37 | ad2antrr 479 | . . . . 5 TopOn |
39 | simprr 521 | . . . . 5 | |
40 | 35, 38, 39 | cnmptc 12451 | . . . 4 |
41 | 35 | cnmptid 12450 | . . . 4 |
42 | 35, 40, 41 | cnmpt1t 12454 | . . 3 |
43 | cnima 12389 | . . 3 | |
44 | 42, 6, 43 | syl2anc 408 | . 2 |
45 | 32, 44 | eqeltrd 2216 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 crab 2420 cvv 2686 wss 3071 csn 3527 cop 3530 cuni 3736 cmpt 3989 cxp 4537 ccnv 4538 cima 4542 cfv 5123 (class class class)co 5774 ctop 12164 TopOnctopon 12177 ccn 12354 ctx 12421 |
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-nul 3364 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-1st 6038 df-2nd 6039 df-map 6544 df-topgen 12141 df-top 12165 df-topon 12178 df-bases 12210 df-cn 12357 df-cnp 12358 df-tx 12422 |
This theorem is referenced by: (None) |
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