Theorem mrcidb 17521

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 17519	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ∈ 𝐶) → (𝐹‘𝑈) = 𝑈)
3		simpr 484	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → (𝐹‘𝑈) = 𝑈)
4	1	mrcssv 17520	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑈) ⊆ 𝑋)
5	4	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → (𝐹‘𝑈) ⊆ 𝑋)
6	3, 5	eqsstrrd 3970	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → 𝑈 ⊆ 𝑋)
7	1	mrccl 17517	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) ∈ 𝐶)
8	6, 7	syldan 591	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → (𝐹‘𝑈) ∈ 𝐶)
9	3, 8	eqeltrrd 2832	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → 𝑈 ∈ 𝐶)
10	2, 9	impbida 800	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) = 𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ⊆ wss 3902  ‘cfv 6481  Moorecmre 17484  mrClscmrc 17485

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-mre 17488  df-mrc 17489

This theorem is referenced by:  mrcidb2  17524