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Mirrors > Home > ILE Home > Th. List > toptopon | GIF version |
Description: Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
toptopon.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
toptopon | ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toptopon.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
2 | istopon 12219 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) | |
3 | 1, 2 | mpbiran2 926 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ Top) |
4 | 3 | bicomi 131 | 1 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1332 ∈ wcel 1481 ∪ cuni 3744 ‘cfv 5131 Topctop 12203 TopOnctopon 12216 |
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-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-topon 12217 |
This theorem is referenced by: toptopon2 12225 eltpsi 12247 restuni 12380 stoig 12381 iscn2 12408 lmcvg 12425 cnpnei 12427 cnss1 12434 cnss2 12435 cncnpi 12436 cncnp2m 12439 cnnei 12440 cnrest 12443 cnrest2 12444 cnrest2r 12445 cnptoprest 12447 cnptoprest2 12448 lmss 12454 txuni 12471 txcnmpt 12481 txcn 12483 cnmpt11 12491 cnmpt11f 12492 imasnopn 12507 hmeof1o 12517 hmeores 12523 txhmeo 12527 retopon 12734 |
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