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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 12453 | . . 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 1335 ∈ wcel 2128 ∪ cuni 3773 ‘cfv 5171 Topctop 12437 TopOnctopon 12450 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-iota 5136 df-fun 5173 df-fv 5179 df-topon 12451 |
This theorem is referenced by: toptopon2 12459 eltpsi 12481 restuni 12614 stoig 12615 iscn2 12642 lmcvg 12659 cnpnei 12661 cnss1 12668 cnss2 12669 cncnpi 12670 cncnp2m 12673 cnnei 12674 cnrest 12677 cnrest2 12678 cnrest2r 12679 cnptoprest 12681 cnptoprest2 12682 lmss 12688 txuni 12705 txcnmpt 12715 txcn 12717 cnmpt11 12725 cnmpt11f 12726 imasnopn 12741 hmeof1o 12751 hmeores 12757 txhmeo 12761 retopon 12968 |
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