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Mirrors > Home > ILE Home > Th. List > bastop2 | GIF version |
Description: A version of bastop1 12798 that doesn't have 𝐵 ⊆ 𝐽 in the antecedent. (Contributed by NM, 3-Feb-2008.) |
Ref | Expression |
---|---|
bastop2 | ⊢ (𝐽 ∈ Top → ((topGen‘𝐵) = 𝐽 ↔ (𝐵 ⊆ 𝐽 ∧ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2233 | . . . . . . . 8 ⊢ ((topGen‘𝐵) = 𝐽 → ((topGen‘𝐵) ∈ Top ↔ 𝐽 ∈ Top)) | |
2 | 1 | biimparc 297 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐵) = 𝐽) → (topGen‘𝐵) ∈ Top) |
3 | tgclb 12780 | . . . . . . 7 ⊢ (𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top) | |
4 | 2, 3 | sylibr 133 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐵) = 𝐽) → 𝐵 ∈ TopBases) |
5 | bastg 12776 | . . . . . 6 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵)) | |
6 | 4, 5 | syl 14 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐵) = 𝐽) → 𝐵 ⊆ (topGen‘𝐵)) |
7 | simpr 109 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐵) = 𝐽) → (topGen‘𝐵) = 𝐽) | |
8 | 6, 7 | sseqtrd 3185 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (topGen‘𝐵) = 𝐽) → 𝐵 ⊆ 𝐽) |
9 | 8 | ex 114 | . . 3 ⊢ (𝐽 ∈ Top → ((topGen‘𝐵) = 𝐽 → 𝐵 ⊆ 𝐽)) |
10 | 9 | pm4.71rd 392 | . 2 ⊢ (𝐽 ∈ Top → ((topGen‘𝐵) = 𝐽 ↔ (𝐵 ⊆ 𝐽 ∧ (topGen‘𝐵) = 𝐽))) |
11 | bastop1 12798 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝐽) → ((topGen‘𝐵) = 𝐽 ↔ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))) | |
12 | 11 | pm5.32da 449 | . 2 ⊢ (𝐽 ∈ Top → ((𝐵 ⊆ 𝐽 ∧ (topGen‘𝐵) = 𝐽) ↔ (𝐵 ⊆ 𝐽 ∧ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))) |
13 | 10, 12 | bitrd 187 | 1 ⊢ (𝐽 ∈ Top → ((topGen‘𝐵) = 𝐽 ↔ (𝐵 ⊆ 𝐽 ∧ ∀𝑥 ∈ 𝐽 ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 ∃wex 1485 ∈ wcel 2141 ∀wral 2448 ⊆ wss 3121 ∪ cuni 3794 ‘cfv 5196 topGenctg 12580 Topctop 12710 TopBasesctb 12755 |
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-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-topgen 12586 df-top 12711 df-bases 12756 |
This theorem is referenced by: (None) |
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