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Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem3 | Structured version Visualization version GIF version |
Description: Lemma for kur14 30452. 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 6104 | . 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 20673 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋) |
7 | 3, 4, 6 | mp2an 704 | . 2 ⊢ ((cls‘𝐽)‘𝐴) ⊆ 𝑋 |
8 | 2, 7 | eqsstri 3598 | 1 ⊢ (𝐾‘𝐴) ⊆ 𝑋 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ∪ cuni 4372 ‘cfv 5804 Topctop 20517 intcnt 20631 clsccl 20632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-top 20521 df-cld 20633 df-cls 20635 |
This theorem is referenced by: kur14lem6 30447 kur14lem7 30448 |
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