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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 1515 | . . . 4 | |
2 | nfcv 2306 | . . . 4 | |
3 | nfrab1 2643 | . . . 4 | |
4 | txtop 12807 | . . . . . . . . . . . . 13 | |
5 | 4 | adantr 274 | . . . . . . . . . . . 12 |
6 | simprl 521 | . . . . . . . . . . . 12 | |
7 | eqid 2164 | . . . . . . . . . . . . 13 | |
8 | 7 | eltopss 12554 | . . . . . . . . . . . 12 |
9 | 5, 6, 8 | syl2anc 409 | . . . . . . . . . . 11 |
10 | imasnopn.1 | . . . . . . . . . . . . 13 | |
11 | eqid 2164 | . . . . . . . . . . . . 13 | |
12 | 10, 11 | txuni 12810 | . . . . . . . . . . . 12 |
13 | 12 | adantr 274 | . . . . . . . . . . 11 |
14 | 9, 13 | sseqtrrd 3176 | . . . . . . . . . 10 |
15 | imass1 4973 | . . . . . . . . . 10 | |
16 | 14, 15 | syl 14 | . . . . . . . . 9 |
17 | xpimasn 5046 | . . . . . . . . . 10 | |
18 | 17 | ad2antll 483 | . . . . . . . . 9 |
19 | 16, 18 | sseqtrd 3175 | . . . . . . . 8 |
20 | 19 | sseld 3136 | . . . . . . 7 |
21 | 20 | pm4.71rd 392 | . . . . . 6 |
22 | elimasng 4966 | . . . . . . . . 9 | |
23 | 22 | elvd 2726 | . . . . . . . 8 |
24 | 23 | ad2antll 483 | . . . . . . 7 |
25 | 24 | anbi2d 460 | . . . . . 6 |
26 | 21, 25 | bitrd 187 | . . . . 5 |
27 | rabid 2639 | . . . . 5 | |
28 | 26, 27 | bitr4di 197 | . . . 4 |
29 | 1, 2, 3, 28 | eqrd 3155 | . . 3 |
30 | eqid 2164 | . . . 4 | |
31 | 30 | mptpreima 5091 | . . 3 |
32 | 29, 31 | eqtr4di 2215 | . 2 |
33 | 11 | toptopon 12563 | . . . . . 6 TopOn |
34 | 33 | biimpi 119 | . . . . 5 TopOn |
35 | 34 | ad2antlr 481 | . . . 4 TopOn |
36 | 10 | toptopon 12563 | . . . . . . 7 TopOn |
37 | 36 | biimpi 119 | . . . . . 6 TopOn |
38 | 37 | ad2antrr 480 | . . . . 5 TopOn |
39 | simprr 522 | . . . . 5 | |
40 | 35, 38, 39 | cnmptc 12829 | . . . 4 |
41 | 35 | cnmptid 12828 | . . . 4 |
42 | 35, 40, 41 | cnmpt1t 12832 | . . 3 |
43 | cnima 12767 | . . 3 | |
44 | 42, 6, 43 | syl2anc 409 | . 2 |
45 | 32, 44 | eqeltrd 2241 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 crab 2446 cvv 2721 wss 3111 csn 3570 cop 3573 cuni 3783 cmpt 4037 cxp 4596 ccnv 4597 cima 4601 cfv 5182 (class class class)co 5836 ctop 12542 TopOnctopon 12555 ccn 12732 ctx 12799 |
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-nul 3405 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-1st 6100 df-2nd 6101 df-map 6607 df-topgen 12519 df-top 12543 df-topon 12556 df-bases 12588 df-cn 12735 df-cnp 12736 df-tx 12800 |
This theorem is referenced by: (None) |
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