Theorem mrcidb 17576

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 17574	. 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 17575	. . . . . 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 3982	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → 𝑈 ⊆ 𝑋)
7	1	mrccl 17572	. . . 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 2829	. 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 1540   ∈ wcel 2109   ⊆ wss 3914  ‘cfv 6511  Moorecmre 17543  mrClscmrc 17544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-mre 17547  df-mrc 17548

This theorem is referenced by:  mrcidb2  17579