Theorem mrcidb 17821

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 17819	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ∈ 𝐶) → (𝐹‘𝑈) = 𝑈)
3		simpr 490	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → (𝐹‘𝑈) = 𝑈)
4	1	mrcssv 17820	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑈) ⊆ 𝑋)
5	4	adantr 486	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → (𝐹‘𝑈) ⊆ 𝑋)
6	3, 5	eqsstrrd 3948	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → 𝑈 ⊆ 𝑋)
7	1	mrccl 17817	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝑋) → (𝐹‘𝑈) ∈ 𝐶)
8	6, 7	syldan 603	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → (𝐹‘𝑈) ∈ 𝐶)
9	3, 8	eqeltrrd 2865	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹‘𝑈) = 𝑈) → 𝑈 ∈ 𝐶)
10	2, 9	impbida 813	1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝐶 ↔ (𝐹‘𝑈) = 𝑈))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  ‘cfv 6561  Moorecmre 17784  mrClscmrc 17785

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-mre 17788  df-mrc 17789

This theorem is used by:  mrcidb2  17824