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Mirrors > Home > ILE Home > Th. List > toponmax | GIF version |
Description: The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
toponmax | ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toponuni 12554 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | |
2 | topontop 12553 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | |
3 | eqid 2164 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | topopn 12547 | . . 3 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 2, 4 | syl 14 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → ∪ 𝐽 ∈ 𝐽) |
6 | 1, 5 | eqeltrd 2241 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 ∪ cuni 3783 ‘cfv 5182 Topctop 12536 TopOnctopon 12549 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-top 12537 df-topon 12550 |
This theorem is referenced by: topgele 12568 eltpsg 12579 resttopon 12712 lmfval 12733 cnfval 12735 cnpfval 12736 iscn 12738 cnpval 12739 iscnp 12740 lmbrf 12756 cnconst2 12774 cnrest2 12777 cndis 12782 cnpdis 12783 lmfss 12785 lmres 12789 lmff 12790 tx1cn 12810 tx2cn 12811 txlm 12820 cnmpt2res 12838 mopnm 12989 isxms2 12993 |
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