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Mirrors > Home > ILE Home > Th. List > idcn | GIF version |
Description: A restricted identity function is a continuous function. (Contributed by FL, 27-Dec-2006.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
idcn | β’ (π½ β (TopOnβπ) β ( I βΎ π) β (π½ Cn π½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3175 | . 2 β’ π½ β π½ | |
2 | ssidcn 13680 | . . 3 β’ ((π½ β (TopOnβπ) β§ π½ β (TopOnβπ)) β (( I βΎ π) β (π½ Cn π½) β π½ β π½)) | |
3 | 2 | anidms 397 | . 2 β’ (π½ β (TopOnβπ) β (( I βΎ π) β (π½ Cn π½) β π½ β π½)) |
4 | 1, 3 | mpbiri 168 | 1 β’ (π½ β (TopOnβπ) β ( I βΎ π) β (π½ Cn π½)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β wb 105 β wcel 2148 β wss 3129 I cid 4288 βΎ cres 4628 βcfv 5216 (class class class)co 5874 TopOnctopon 13480 Cn ccn 13655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-map 6649 df-top 13468 df-topon 13481 df-cn 13658 |
This theorem is referenced by: cnmptid 13751 idhmeo 13787 |
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