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Theorem mrcuni 16328
 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 472 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 simpll 805 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → 𝐶 ∈ (Moore‘𝑋))
3 ssel2 3631 . . . . . . . . 9 ((𝑈 ⊆ 𝒫 𝑋𝑠𝑈) → 𝑠 ∈ 𝒫 𝑋)
43elpwid 4203 . . . . . . . 8 ((𝑈 ⊆ 𝒫 𝑋𝑠𝑈) → 𝑠𝑋)
54adantll 750 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → 𝑠𝑋)
6 mrcfval.f . . . . . . . 8 𝐹 = (mrCls‘𝐶)
76mrcssid 16324 . . . . . . 7 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠𝑋) → 𝑠 ⊆ (𝐹𝑠))
82, 5, 7syl2anc 694 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → 𝑠 ⊆ (𝐹𝑠))
96mrcf 16316 . . . . . . . . . . 11 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
10 ffun 6086 . . . . . . . . . . 11 (𝐹:𝒫 𝑋𝐶 → Fun 𝐹)
119, 10syl 17 . . . . . . . . . 10 (𝐶 ∈ (Moore‘𝑋) → Fun 𝐹)
1211adantr 480 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → Fun 𝐹)
13 fdm 6089 . . . . . . . . . . . 12 (𝐹:𝒫 𝑋𝐶 → dom 𝐹 = 𝒫 𝑋)
149, 13syl 17 . . . . . . . . . . 11 (𝐶 ∈ (Moore‘𝑋) → dom 𝐹 = 𝒫 𝑋)
1514sseq2d 3666 . . . . . . . . . 10 (𝐶 ∈ (Moore‘𝑋) → (𝑈 ⊆ dom 𝐹𝑈 ⊆ 𝒫 𝑋))
1615biimpar 501 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝑈 ⊆ dom 𝐹)
17 funfvima2 6533 . . . . . . . . 9 ((Fun 𝐹𝑈 ⊆ dom 𝐹) → (𝑠𝑈 → (𝐹𝑠) ∈ (𝐹𝑈)))
1812, 16, 17syl2anc 694 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝑠𝑈 → (𝐹𝑠) ∈ (𝐹𝑈)))
1918imp 444 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → (𝐹𝑠) ∈ (𝐹𝑈))
20 elssuni 4499 . . . . . . 7 ((𝐹𝑠) ∈ (𝐹𝑈) → (𝐹𝑠) ⊆ (𝐹𝑈))
2119, 20syl 17 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → (𝐹𝑠) ⊆ (𝐹𝑈))
228, 21sstrd 3646 . . . . 5 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → 𝑠 (𝐹𝑈))
2322ralrimiva 2995 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠𝑈 𝑠 (𝐹𝑈))
24 unissb 4501 . . . 4 ( 𝑈 (𝐹𝑈) ↔ ∀𝑠𝑈 𝑠 (𝐹𝑈))
2523, 24sylibr 224 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝑈 (𝐹𝑈))
266mrcssv 16321 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑥) ⊆ 𝑋)
2726adantr 480 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹𝑥) ⊆ 𝑋)
2827ralrimivw 2996 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑥𝑈 (𝐹𝑥) ⊆ 𝑋)
29 ffn 6083 . . . . . . 7 (𝐹:𝒫 𝑋𝐶𝐹 Fn 𝒫 𝑋)
309, 29syl 17 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
31 sseq1 3659 . . . . . . 7 (𝑠 = (𝐹𝑥) → (𝑠𝑋 ↔ (𝐹𝑥) ⊆ 𝑋))
3231ralima 6538 . . . . . 6 ((𝐹 Fn 𝒫 𝑋𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹𝑈)𝑠𝑋 ↔ ∀𝑥𝑈 (𝐹𝑥) ⊆ 𝑋))
3330, 32sylan 487 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹𝑈)𝑠𝑋 ↔ ∀𝑥𝑈 (𝐹𝑥) ⊆ 𝑋))
3428, 33mpbird 247 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠 ∈ (𝐹𝑈)𝑠𝑋)
35 unissb 4501 . . . 4 ( (𝐹𝑈) ⊆ 𝑋 ↔ ∀𝑠 ∈ (𝐹𝑈)𝑠𝑋)
3634, 35sylibr 224 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹𝑈) ⊆ 𝑋)
376mrcss 16323 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 (𝐹𝑈) ∧ (𝐹𝑈) ⊆ 𝑋) → (𝐹 𝑈) ⊆ (𝐹 (𝐹𝑈)))
381, 25, 36, 37syl3anc 1366 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 𝑈) ⊆ (𝐹 (𝐹𝑈)))
39 simpll 805 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥𝑈) → 𝐶 ∈ (Moore‘𝑋))
40 elssuni 4499 . . . . . . . . 9 (𝑥𝑈𝑥 𝑈)
4140adantl 481 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥𝑈) → 𝑥 𝑈)
42 sspwuni 4643 . . . . . . . . . . 11 (𝑈 ⊆ 𝒫 𝑋 𝑈𝑋)
4342biimpi 206 . . . . . . . . . 10 (𝑈 ⊆ 𝒫 𝑋 𝑈𝑋)
4443adantl 481 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝑈𝑋)
4544adantr 480 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥𝑈) → 𝑈𝑋)
466mrcss 16323 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 𝑈 𝑈𝑋) → (𝐹𝑥) ⊆ (𝐹 𝑈))
4739, 41, 45, 46syl3anc 1366 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥𝑈) → (𝐹𝑥) ⊆ (𝐹 𝑈))
4847ralrimiva 2995 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑥𝑈 (𝐹𝑥) ⊆ (𝐹 𝑈))
49 sseq1 3659 . . . . . . . 8 (𝑠 = (𝐹𝑥) → (𝑠 ⊆ (𝐹 𝑈) ↔ (𝐹𝑥) ⊆ (𝐹 𝑈)))
5049ralima 6538 . . . . . . 7 ((𝐹 Fn 𝒫 𝑋𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹𝑈)𝑠 ⊆ (𝐹 𝑈) ↔ ∀𝑥𝑈 (𝐹𝑥) ⊆ (𝐹 𝑈)))
5130, 50sylan 487 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹𝑈)𝑠 ⊆ (𝐹 𝑈) ↔ ∀𝑥𝑈 (𝐹𝑥) ⊆ (𝐹 𝑈)))
5248, 51mpbird 247 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠 ∈ (𝐹𝑈)𝑠 ⊆ (𝐹 𝑈))
53 unissb 4501 . . . . 5 ( (𝐹𝑈) ⊆ (𝐹 𝑈) ↔ ∀𝑠 ∈ (𝐹𝑈)𝑠 ⊆ (𝐹 𝑈))
5452, 53sylibr 224 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹𝑈) ⊆ (𝐹 𝑈))
556mrcssv 16321 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → (𝐹 𝑈) ⊆ 𝑋)
5655adantr 480 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 𝑈) ⊆ 𝑋)
576mrcss 16323 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) ⊆ (𝐹 𝑈) ∧ (𝐹 𝑈) ⊆ 𝑋) → (𝐹 (𝐹𝑈)) ⊆ (𝐹‘(𝐹 𝑈)))
581, 54, 56, 57syl3anc 1366 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 (𝐹𝑈)) ⊆ (𝐹‘(𝐹 𝑈)))
596mrcidm 16326 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹‘(𝐹 𝑈)) = (𝐹 𝑈))
601, 44, 59syl2anc 694 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘(𝐹 𝑈)) = (𝐹 𝑈))
6158, 60sseqtrd 3674 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 (𝐹𝑈)) ⊆ (𝐹 𝑈))
6238, 61eqssd 3653 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 𝑈) = (𝐹 (𝐹𝑈)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941   ⊆ wss 3607  𝒫 cpw 4191  ∪ cuni 4468  dom cdm 5143   “ cima 5146  Fun wfun 5920   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  Moorecmre 16289  mrClscmrc 16290 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-mre 16293  df-mrc 16294 This theorem is referenced by:  mrcun  16329  isacs4lem  17215
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