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Theorem mrcuni 16884
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 486 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 simpll 766 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → 𝐶 ∈ (Moore‘𝑋))
3 ssel2 3910 . . . . . . . . 9 ((𝑈 ⊆ 𝒫 𝑋𝑠𝑈) → 𝑠 ∈ 𝒫 𝑋)
43elpwid 4508 . . . . . . . 8 ((𝑈 ⊆ 𝒫 𝑋𝑠𝑈) → 𝑠𝑋)
54adantll 713 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → 𝑠𝑋)
6 mrcfval.f . . . . . . . 8 𝐹 = (mrCls‘𝐶)
76mrcssid 16880 . . . . . . 7 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠𝑋) → 𝑠 ⊆ (𝐹𝑠))
82, 5, 7syl2anc 587 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → 𝑠 ⊆ (𝐹𝑠))
96mrcf 16872 . . . . . . . . . . 11 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
109ffund 6491 . . . . . . . . . 10 (𝐶 ∈ (Moore‘𝑋) → Fun 𝐹)
1110adantr 484 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → Fun 𝐹)
129fdmd 6497 . . . . . . . . . . 11 (𝐶 ∈ (Moore‘𝑋) → dom 𝐹 = 𝒫 𝑋)
1312sseq2d 3947 . . . . . . . . . 10 (𝐶 ∈ (Moore‘𝑋) → (𝑈 ⊆ dom 𝐹𝑈 ⊆ 𝒫 𝑋))
1413biimpar 481 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝑈 ⊆ dom 𝐹)
15 funfvima2 6971 . . . . . . . . 9 ((Fun 𝐹𝑈 ⊆ dom 𝐹) → (𝑠𝑈 → (𝐹𝑠) ∈ (𝐹𝑈)))
1611, 14, 15syl2anc 587 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝑠𝑈 → (𝐹𝑠) ∈ (𝐹𝑈)))
1716imp 410 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → (𝐹𝑠) ∈ (𝐹𝑈))
18 elssuni 4830 . . . . . . 7 ((𝐹𝑠) ∈ (𝐹𝑈) → (𝐹𝑠) ⊆ (𝐹𝑈))
1917, 18syl 17 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → (𝐹𝑠) ⊆ (𝐹𝑈))
208, 19sstrd 3925 . . . . 5 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → 𝑠 (𝐹𝑈))
2120ralrimiva 3149 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠𝑈 𝑠 (𝐹𝑈))
22 unissb 4832 . . . 4 ( 𝑈 (𝐹𝑈) ↔ ∀𝑠𝑈 𝑠 (𝐹𝑈))
2321, 22sylibr 237 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝑈 (𝐹𝑈))
246mrcssv 16877 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑥) ⊆ 𝑋)
2524adantr 484 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹𝑥) ⊆ 𝑋)
2625ralrimivw 3150 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑥𝑈 (𝐹𝑥) ⊆ 𝑋)
279ffnd 6488 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
28 sseq1 3940 . . . . . . 7 (𝑠 = (𝐹𝑥) → (𝑠𝑋 ↔ (𝐹𝑥) ⊆ 𝑋))
2928ralima 6978 . . . . . 6 ((𝐹 Fn 𝒫 𝑋𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹𝑈)𝑠𝑋 ↔ ∀𝑥𝑈 (𝐹𝑥) ⊆ 𝑋))
3027, 29sylan 583 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹𝑈)𝑠𝑋 ↔ ∀𝑥𝑈 (𝐹𝑥) ⊆ 𝑋))
3126, 30mpbird 260 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠 ∈ (𝐹𝑈)𝑠𝑋)
32 unissb 4832 . . . 4 ( (𝐹𝑈) ⊆ 𝑋 ↔ ∀𝑠 ∈ (𝐹𝑈)𝑠𝑋)
3331, 32sylibr 237 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹𝑈) ⊆ 𝑋)
346mrcss 16879 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 (𝐹𝑈) ∧ (𝐹𝑈) ⊆ 𝑋) → (𝐹 𝑈) ⊆ (𝐹 (𝐹𝑈)))
351, 23, 33, 34syl3anc 1368 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 𝑈) ⊆ (𝐹 (𝐹𝑈)))
36 simpll 766 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥𝑈) → 𝐶 ∈ (Moore‘𝑋))
37 elssuni 4830 . . . . . . . . 9 (𝑥𝑈𝑥 𝑈)
3837adantl 485 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥𝑈) → 𝑥 𝑈)
39 sspwuni 4985 . . . . . . . . . . 11 (𝑈 ⊆ 𝒫 𝑋 𝑈𝑋)
4039biimpi 219 . . . . . . . . . 10 (𝑈 ⊆ 𝒫 𝑋 𝑈𝑋)
4140adantl 485 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝑈𝑋)
4241adantr 484 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥𝑈) → 𝑈𝑋)
436mrcss 16879 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 𝑈 𝑈𝑋) → (𝐹𝑥) ⊆ (𝐹 𝑈))
4436, 38, 42, 43syl3anc 1368 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥𝑈) → (𝐹𝑥) ⊆ (𝐹 𝑈))
4544ralrimiva 3149 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑥𝑈 (𝐹𝑥) ⊆ (𝐹 𝑈))
46 sseq1 3940 . . . . . . . 8 (𝑠 = (𝐹𝑥) → (𝑠 ⊆ (𝐹 𝑈) ↔ (𝐹𝑥) ⊆ (𝐹 𝑈)))
4746ralima 6978 . . . . . . 7 ((𝐹 Fn 𝒫 𝑋𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹𝑈)𝑠 ⊆ (𝐹 𝑈) ↔ ∀𝑥𝑈 (𝐹𝑥) ⊆ (𝐹 𝑈)))
4827, 47sylan 583 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹𝑈)𝑠 ⊆ (𝐹 𝑈) ↔ ∀𝑥𝑈 (𝐹𝑥) ⊆ (𝐹 𝑈)))
4945, 48mpbird 260 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠 ∈ (𝐹𝑈)𝑠 ⊆ (𝐹 𝑈))
50 unissb 4832 . . . . 5 ( (𝐹𝑈) ⊆ (𝐹 𝑈) ↔ ∀𝑠 ∈ (𝐹𝑈)𝑠 ⊆ (𝐹 𝑈))
5149, 50sylibr 237 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹𝑈) ⊆ (𝐹 𝑈))
526mrcssv 16877 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → (𝐹 𝑈) ⊆ 𝑋)
5352adantr 484 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 𝑈) ⊆ 𝑋)
546mrcss 16879 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) ⊆ (𝐹 𝑈) ∧ (𝐹 𝑈) ⊆ 𝑋) → (𝐹 (𝐹𝑈)) ⊆ (𝐹‘(𝐹 𝑈)))
551, 51, 53, 54syl3anc 1368 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 (𝐹𝑈)) ⊆ (𝐹‘(𝐹 𝑈)))
566mrcidm 16882 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹‘(𝐹 𝑈)) = (𝐹 𝑈))
571, 41, 56syl2anc 587 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘(𝐹 𝑈)) = (𝐹 𝑈))
5855, 57sseqtrd 3955 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 (𝐹𝑈)) ⊆ (𝐹 𝑈))
5935, 58eqssd 3932 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 𝑈) = (𝐹 (𝐹𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  wral 3106  wss 3881  𝒫 cpw 4497   cuni 4800  dom cdm 5519  cima 5522  Fun wfun 6318   Fn wfn 6319  cfv 6324  Moorecmre 16845  mrClscmrc 16846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-mre 16849  df-mrc 16850
This theorem is referenced by:  mrcun  16885  isacs4lem  17770
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