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Theorem mrcuni 16880
Description: Idempotence of closure under a general union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f 𝐹 = (mrCls‘𝐶)
Assertion
Ref Expression
mrcuni ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 𝑈) = (𝐹 (𝐹𝑈)))

Proof of Theorem mrcuni
Dummy variables 𝑥 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 483 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 simpll 763 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → 𝐶 ∈ (Moore‘𝑋))
3 ssel2 3959 . . . . . . . . 9 ((𝑈 ⊆ 𝒫 𝑋𝑠𝑈) → 𝑠 ∈ 𝒫 𝑋)
43elpwid 4549 . . . . . . . 8 ((𝑈 ⊆ 𝒫 𝑋𝑠𝑈) → 𝑠𝑋)
54adantll 710 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → 𝑠𝑋)
6 mrcfval.f . . . . . . . 8 𝐹 = (mrCls‘𝐶)
76mrcssid 16876 . . . . . . 7 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠𝑋) → 𝑠 ⊆ (𝐹𝑠))
82, 5, 7syl2anc 584 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → 𝑠 ⊆ (𝐹𝑠))
96mrcf 16868 . . . . . . . . . . 11 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
109ffund 6511 . . . . . . . . . 10 (𝐶 ∈ (Moore‘𝑋) → Fun 𝐹)
1110adantr 481 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → Fun 𝐹)
129fdmd 6516 . . . . . . . . . . 11 (𝐶 ∈ (Moore‘𝑋) → dom 𝐹 = 𝒫 𝑋)
1312sseq2d 3996 . . . . . . . . . 10 (𝐶 ∈ (Moore‘𝑋) → (𝑈 ⊆ dom 𝐹𝑈 ⊆ 𝒫 𝑋))
1413biimpar 478 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝑈 ⊆ dom 𝐹)
15 funfvima2 6984 . . . . . . . . 9 ((Fun 𝐹𝑈 ⊆ dom 𝐹) → (𝑠𝑈 → (𝐹𝑠) ∈ (𝐹𝑈)))
1611, 14, 15syl2anc 584 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝑠𝑈 → (𝐹𝑠) ∈ (𝐹𝑈)))
1716imp 407 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → (𝐹𝑠) ∈ (𝐹𝑈))
18 elssuni 4859 . . . . . . 7 ((𝐹𝑠) ∈ (𝐹𝑈) → (𝐹𝑠) ⊆ (𝐹𝑈))
1917, 18syl 17 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → (𝐹𝑠) ⊆ (𝐹𝑈))
208, 19sstrd 3974 . . . . 5 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → 𝑠 (𝐹𝑈))
2120ralrimiva 3179 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠𝑈 𝑠 (𝐹𝑈))
22 unissb 4861 . . . 4 ( 𝑈 (𝐹𝑈) ↔ ∀𝑠𝑈 𝑠 (𝐹𝑈))
2321, 22sylibr 235 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝑈 (𝐹𝑈))
246mrcssv 16873 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑥) ⊆ 𝑋)
2524adantr 481 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹𝑥) ⊆ 𝑋)
2625ralrimivw 3180 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑥𝑈 (𝐹𝑥) ⊆ 𝑋)
279ffnd 6508 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
28 sseq1 3989 . . . . . . 7 (𝑠 = (𝐹𝑥) → (𝑠𝑋 ↔ (𝐹𝑥) ⊆ 𝑋))
2928ralima 6991 . . . . . 6 ((𝐹 Fn 𝒫 𝑋𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹𝑈)𝑠𝑋 ↔ ∀𝑥𝑈 (𝐹𝑥) ⊆ 𝑋))
3027, 29sylan 580 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹𝑈)𝑠𝑋 ↔ ∀𝑥𝑈 (𝐹𝑥) ⊆ 𝑋))
3126, 30mpbird 258 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠 ∈ (𝐹𝑈)𝑠𝑋)
32 unissb 4861 . . . 4 ( (𝐹𝑈) ⊆ 𝑋 ↔ ∀𝑠 ∈ (𝐹𝑈)𝑠𝑋)
3331, 32sylibr 235 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹𝑈) ⊆ 𝑋)
346mrcss 16875 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 (𝐹𝑈) ∧ (𝐹𝑈) ⊆ 𝑋) → (𝐹 𝑈) ⊆ (𝐹 (𝐹𝑈)))
351, 23, 33, 34syl3anc 1363 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 𝑈) ⊆ (𝐹 (𝐹𝑈)))
36 simpll 763 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥𝑈) → 𝐶 ∈ (Moore‘𝑋))
37 elssuni 4859 . . . . . . . . 9 (𝑥𝑈𝑥 𝑈)
3837adantl 482 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥𝑈) → 𝑥 𝑈)
39 sspwuni 5013 . . . . . . . . . . 11 (𝑈 ⊆ 𝒫 𝑋 𝑈𝑋)
4039biimpi 217 . . . . . . . . . 10 (𝑈 ⊆ 𝒫 𝑋 𝑈𝑋)
4140adantl 482 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝑈𝑋)
4241adantr 481 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥𝑈) → 𝑈𝑋)
436mrcss 16875 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 𝑈 𝑈𝑋) → (𝐹𝑥) ⊆ (𝐹 𝑈))
4436, 38, 42, 43syl3anc 1363 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥𝑈) → (𝐹𝑥) ⊆ (𝐹 𝑈))
4544ralrimiva 3179 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑥𝑈 (𝐹𝑥) ⊆ (𝐹 𝑈))
46 sseq1 3989 . . . . . . . 8 (𝑠 = (𝐹𝑥) → (𝑠 ⊆ (𝐹 𝑈) ↔ (𝐹𝑥) ⊆ (𝐹 𝑈)))
4746ralima 6991 . . . . . . 7 ((𝐹 Fn 𝒫 𝑋𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹𝑈)𝑠 ⊆ (𝐹 𝑈) ↔ ∀𝑥𝑈 (𝐹𝑥) ⊆ (𝐹 𝑈)))
4827, 47sylan 580 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹𝑈)𝑠 ⊆ (𝐹 𝑈) ↔ ∀𝑥𝑈 (𝐹𝑥) ⊆ (𝐹 𝑈)))
4945, 48mpbird 258 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠 ∈ (𝐹𝑈)𝑠 ⊆ (𝐹 𝑈))
50 unissb 4861 . . . . 5 ( (𝐹𝑈) ⊆ (𝐹 𝑈) ↔ ∀𝑠 ∈ (𝐹𝑈)𝑠 ⊆ (𝐹 𝑈))
5149, 50sylibr 235 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹𝑈) ⊆ (𝐹 𝑈))
526mrcssv 16873 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → (𝐹 𝑈) ⊆ 𝑋)
5352adantr 481 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 𝑈) ⊆ 𝑋)
546mrcss 16875 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) ⊆ (𝐹 𝑈) ∧ (𝐹 𝑈) ⊆ 𝑋) → (𝐹 (𝐹𝑈)) ⊆ (𝐹‘(𝐹 𝑈)))
551, 51, 53, 54syl3anc 1363 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 (𝐹𝑈)) ⊆ (𝐹‘(𝐹 𝑈)))
566mrcidm 16878 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹‘(𝐹 𝑈)) = (𝐹 𝑈))
571, 41, 56syl2anc 584 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘(𝐹 𝑈)) = (𝐹 𝑈))
5855, 57sseqtrd 4004 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 (𝐹𝑈)) ⊆ (𝐹 𝑈))
5935, 58eqssd 3981 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 𝑈) = (𝐹 (𝐹𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wss 3933  𝒫 cpw 4535   cuni 4830  dom cdm 5548  cima 5551  Fun wfun 6342   Fn wfn 6343  cfv 6348  Moorecmre 16841  mrClscmrc 16842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-mre 16845  df-mrc 16846
This theorem is referenced by:  mrcun  16881  isacs4lem  17766
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