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Mirrors > Home > MPE Home > Th. List > mrcidb2 | Structured version Visualization version GIF version |
Description: A set is closed iff it contains its closure. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
Ref | Expression |
---|---|
mrcfval.f | ⊢ 𝐹 = (mrCls‘𝐶) |
Ref | Expression |
---|---|
mrcidb2 | ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) ⊆ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrcfval.f | . . . 4 ⊢ 𝐹 = (mrCls‘𝐶) | |
2 | 1 | mrcidb 16888 | . . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) = 𝑈)) |
3 | 2 | adantr 483 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) = 𝑈)) |
4 | 1 | mrcssid 16890 | . . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → 𝑈 ⊆ (𝐹‘𝑈)) |
5 | 4 | biantrud 534 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → ((𝐹‘𝑈) ⊆ 𝑈 ↔ ((𝐹‘𝑈) ⊆ 𝑈 ∧ 𝑈 ⊆ (𝐹‘𝑈)))) |
6 | eqss 3984 | . . 3 ⊢ ((𝐹‘𝑈) = 𝑈 ↔ ((𝐹‘𝑈) ⊆ 𝑈 ∧ 𝑈 ⊆ (𝐹‘𝑈))) | |
7 | 5, 6 | syl6rbbr 292 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → ((𝐹‘𝑈) = 𝑈 ↔ (𝐹‘𝑈) ⊆ 𝑈)) |
8 | 3, 7 | bitrd 281 | 1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) ⊆ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ‘cfv 6357 Moorecmre 16855 mrClscmrc 16856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-mre 16859 df-mrc 16860 |
This theorem is referenced by: isacs5 17784 |
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