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Theorem mrcidb 16483

Description: A set is closed iff it is equal to its closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)

**Hypothesis**
Ref	Expression
mrcfval.f	⊢ 𝐹 = (mrCls‘𝐶)

**Assertion**
Ref	Expression
mrcidb	⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) = 𝑈))

**Proof of Theorem mrcidb**
Step	Hyp	Ref	Expression
1		mrcfval.f	. . 3 ⊢ 𝐹 = (mrCls‘𝐶)
2	1	mrcid 16481	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ∈ 𝐶) → (𝐹‘𝑈) = 𝑈)
3		simpr 471	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → (𝐹‘𝑈) = 𝑈)
4	1	mrcssv 16482	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑈) ⊆ 𝑋)
5	4	adantr 466	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → (𝐹‘𝑈) ⊆ 𝑋)
6	3, 5	eqsstr3d 3789	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → 𝑈 ⊆ 𝑋)
7	1	mrccl 16479	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) ∈ 𝐶)
8	6, 7	syldan 579	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → (𝐹‘𝑈) ∈ 𝐶)
9	3, 8	eqeltrrd 2851	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → 𝑈 ∈ 𝐶)
10	2, 9	impbida 802	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) = 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ⊆ wss 3723 ‘cfv 6030 Moorecmre 16450 mrClscmrc 16451

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-int 4613 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-fv 6038 df-mre 16454 df-mrc 16455

This theorem is referenced by: mrcidb2 16486