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Mirrors > Home > ILE Home > Th. List > hmeoimaf1o | Unicode version |
Description: The function mapping open sets to their images under a homeomorphism is a bijection of topologies. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
hmeoimaf1o.1 |
Ref | Expression |
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hmeoimaf1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmeoimaf1o.1 | . 2 | |
2 | hmeoima 13104 | . 2 | |
3 | hmeocn 13099 | . . 3 | |
4 | cnima 13014 | . . 3 | |
5 | 3, 4 | sylan 281 | . 2 |
6 | eqid 2170 | . . . . . . 7 | |
7 | eqid 2170 | . . . . . . 7 | |
8 | 6, 7 | hmeof1o 13103 | . . . . . 6 |
9 | 8 | adantr 274 | . . . . 5 |
10 | f1of1 5441 | . . . . 5 | |
11 | 9, 10 | syl 14 | . . . 4 |
12 | elssuni 3824 | . . . . 5 | |
13 | 12 | ad2antrl 487 | . . . 4 |
14 | cnvimass 4974 | . . . . 5 | |
15 | f1dm 5408 | . . . . . 6 | |
16 | 11, 15 | syl 14 | . . . . 5 |
17 | 14, 16 | sseqtrid 3197 | . . . 4 |
18 | f1imaeq 5754 | . . . 4 | |
19 | 11, 13, 17, 18 | syl12anc 1231 | . . 3 |
20 | f1ofo 5449 | . . . . . . 7 | |
21 | 9, 20 | syl 14 | . . . . . 6 |
22 | elssuni 3824 | . . . . . . 7 | |
23 | 22 | ad2antll 488 | . . . . . 6 |
24 | foimacnv 5460 | . . . . . 6 | |
25 | 21, 23, 24 | syl2anc 409 | . . . . 5 |
26 | 25 | eqeq2d 2182 | . . . 4 |
27 | eqcom 2172 | . . . 4 | |
28 | 26, 27 | bitrdi 195 | . . 3 |
29 | 19, 28 | bitr3d 189 | . 2 |
30 | 1, 2, 5, 29 | f1o2d 6054 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wss 3121 cuni 3796 cmpt 4050 ccnv 4610 cdm 4611 cima 4614 wf1 5195 wfo 5196 wf1o 5197 (class class class)co 5853 ccn 12979 chmeo 13094 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-map 6628 df-top 12790 df-topon 12803 df-cn 12982 df-hmeo 13095 |
This theorem is referenced by: (None) |
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