Theorem mrcidb 17673

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 17671	. 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 17672	. . . . . 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 4048	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → 𝑈 ⊆ 𝑋)
7	1	mrccl 17669	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) ∈ 𝐶)
8	6, 7	syldan 590	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → (𝐹‘𝑈) ∈ 𝐶)
9	3, 8	eqeltrrd 2845	. 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 1537   ∈ wcel 2108   ⊆ wss 3976  ‘cfv 6573  Moorecmre 17640  mrClscmrc 17641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-mre 17644  df-mrc 17645

This theorem is referenced by:  mrcidb2  17676