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Mirrors > Home > ILE Home > Th. List > toponcom | GIF version |
Description: If πΎ is a topology on the base set of topology π½, then π½ is a topology on the base of πΎ. (Contributed by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
toponcom | β’ ((π½ β Top β§ πΎ β (TopOnββͺ π½)) β π½ β (TopOnββͺ πΎ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toponuni 13655 | . . . 4 β’ (πΎ β (TopOnββͺ π½) β βͺ π½ = βͺ πΎ) | |
2 | 1 | eqcomd 2183 | . . 3 β’ (πΎ β (TopOnββͺ π½) β βͺ πΎ = βͺ π½) |
3 | 2 | anim2i 342 | . 2 β’ ((π½ β Top β§ πΎ β (TopOnββͺ π½)) β (π½ β Top β§ βͺ πΎ = βͺ π½)) |
4 | istopon 13653 | . 2 β’ (π½ β (TopOnββͺ πΎ) β (π½ β Top β§ βͺ πΎ = βͺ π½)) | |
5 | 3, 4 | sylibr 134 | 1 β’ ((π½ β Top β§ πΎ β (TopOnββͺ π½)) β π½ β (TopOnββͺ πΎ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 βͺ cuni 3811 βcfv 5218 Topctop 13637 TopOnctopon 13650 |
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-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 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 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-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-topon 13651 |
This theorem is referenced by: toponcomb 13668 |
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