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Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem3 | Structured version Visualization version GIF version |
Description: Lemma for kur14 31476. A closure is a subset of the base set. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
kur14lem.j | ⊢ 𝐽 ∈ Top |
kur14lem.x | ⊢ 𝑋 = ∪ 𝐽 |
kur14lem.k | ⊢ 𝐾 = (cls‘𝐽) |
kur14lem.i | ⊢ 𝐼 = (int‘𝐽) |
kur14lem.a | ⊢ 𝐴 ⊆ 𝑋 |
Ref | Expression |
---|---|
kur14lem3 | ⊢ (𝐾‘𝐴) ⊆ 𝑋 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kur14lem.k | . . 3 ⊢ 𝐾 = (cls‘𝐽) | |
2 | 1 | fveq1i 6341 | . 2 ⊢ (𝐾‘𝐴) = ((cls‘𝐽)‘𝐴) |
3 | kur14lem.j | . . 3 ⊢ 𝐽 ∈ Top | |
4 | kur14lem.a | . . 3 ⊢ 𝐴 ⊆ 𝑋 | |
5 | kur14lem.x | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
6 | 5 | clsss3 21036 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋) |
7 | 3, 4, 6 | mp2an 710 | . 2 ⊢ ((cls‘𝐽)‘𝐴) ⊆ 𝑋 |
8 | 2, 7 | eqsstri 3764 | 1 ⊢ (𝐾‘𝐴) ⊆ 𝑋 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1620 ∈ wcel 2127 ⊆ wss 3703 ∪ cuni 4576 ‘cfv 6037 Topctop 20871 intcnt 20994 clsccl 20995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-rep 4911 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-ral 3043 df-rex 3044 df-reu 3045 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-op 4316 df-uni 4577 df-int 4616 df-iun 4662 df-iin 4663 df-br 4793 df-opab 4853 df-mpt 4870 df-id 5162 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-top 20872 df-cld 20996 df-cls 20998 |
This theorem is referenced by: kur14lem6 31471 kur14lem7 31472 |
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