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Theorem mrcuni 16221
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 473 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 simpll 789 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → 𝐶 ∈ (Moore‘𝑋))
3 ssel2 3583 . . . . . . . . 9 ((𝑈 ⊆ 𝒫 𝑋𝑠𝑈) → 𝑠 ∈ 𝒫 𝑋)
43elpwid 4148 . . . . . . . 8 ((𝑈 ⊆ 𝒫 𝑋𝑠𝑈) → 𝑠𝑋)
54adantll 749 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → 𝑠𝑋)
6 mrcfval.f . . . . . . . 8 𝐹 = (mrCls‘𝐶)
76mrcssid 16217 . . . . . . 7 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠𝑋) → 𝑠 ⊆ (𝐹𝑠))
82, 5, 7syl2anc 692 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → 𝑠 ⊆ (𝐹𝑠))
96mrcf 16209 . . . . . . . . . . 11 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
10 ffun 6015 . . . . . . . . . . 11 (𝐹:𝒫 𝑋𝐶 → Fun 𝐹)
119, 10syl 17 . . . . . . . . . 10 (𝐶 ∈ (Moore‘𝑋) → Fun 𝐹)
1211adantr 481 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → Fun 𝐹)
13 fdm 6018 . . . . . . . . . . . 12 (𝐹:𝒫 𝑋𝐶 → dom 𝐹 = 𝒫 𝑋)
149, 13syl 17 . . . . . . . . . . 11 (𝐶 ∈ (Moore‘𝑋) → dom 𝐹 = 𝒫 𝑋)
1514sseq2d 3618 . . . . . . . . . 10 (𝐶 ∈ (Moore‘𝑋) → (𝑈 ⊆ dom 𝐹𝑈 ⊆ 𝒫 𝑋))
1615biimpar 502 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝑈 ⊆ dom 𝐹)
17 funfvima2 6458 . . . . . . . . 9 ((Fun 𝐹𝑈 ⊆ dom 𝐹) → (𝑠𝑈 → (𝐹𝑠) ∈ (𝐹𝑈)))
1812, 16, 17syl2anc 692 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝑠𝑈 → (𝐹𝑠) ∈ (𝐹𝑈)))
1918imp 445 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → (𝐹𝑠) ∈ (𝐹𝑈))
20 elssuni 4440 . . . . . . 7 ((𝐹𝑠) ∈ (𝐹𝑈) → (𝐹𝑠) ⊆ (𝐹𝑈))
2119, 20syl 17 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → (𝐹𝑠) ⊆ (𝐹𝑈))
228, 21sstrd 3598 . . . . 5 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑠𝑈) → 𝑠 (𝐹𝑈))
2322ralrimiva 2962 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑠𝑈 𝑠 (𝐹𝑈))
24 unissb 4442 . . . 4 ( 𝑈 (𝐹𝑈) ↔ ∀𝑠𝑈 𝑠 (𝐹𝑈))
2523, 24sylibr 224 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝑈 (𝐹𝑈))
266mrcssv 16214 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → (𝐹𝑥) ⊆ 𝑋)
2726adantr 481 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹𝑥) ⊆ 𝑋)
2827ralrimivw 2963 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑥𝑈 (𝐹𝑥) ⊆ 𝑋)
29 ffn 6012 . . . . . . 7 (𝐹:𝒫 𝑋𝐶𝐹 Fn 𝒫 𝑋)
309, 29syl 17 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
31 sseq1 3611 . . . . . . 7 (𝑠 = (𝐹𝑥) → (𝑠𝑋 ↔ (𝐹𝑥) ⊆ 𝑋))
3231ralima 6463 . . . . . 6 ((𝐹 Fn 𝒫 𝑋𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹𝑈)𝑠𝑋 ↔ ∀𝑥𝑈 (𝐹𝑥) ⊆ 𝑋))
3330, 32sylan 488 . . . . 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 4442 . . . 4 ( (𝐹𝑈) ⊆ 𝑋 ↔ ∀𝑠 ∈ (𝐹𝑈)𝑠𝑋)
3634, 35sylibr 224 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹𝑈) ⊆ 𝑋)
376mrcss 16216 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 (𝐹𝑈) ∧ (𝐹𝑈) ⊆ 𝑋) → (𝐹 𝑈) ⊆ (𝐹 (𝐹𝑈)))
381, 25, 36, 37syl3anc 1323 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 𝑈) ⊆ (𝐹 (𝐹𝑈)))
39 simpll 789 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥𝑈) → 𝐶 ∈ (Moore‘𝑋))
40 elssuni 4440 . . . . . . . . 9 (𝑥𝑈𝑥 𝑈)
4140adantl 482 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥𝑈) → 𝑥 𝑈)
42 sspwuni 4584 . . . . . . . . . . 11 (𝑈 ⊆ 𝒫 𝑋 𝑈𝑋)
4342biimpi 206 . . . . . . . . . 10 (𝑈 ⊆ 𝒫 𝑋 𝑈𝑋)
4443adantl 482 . . . . . . . . 9 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → 𝑈𝑋)
4544adantr 481 . . . . . . . 8 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥𝑈) → 𝑈𝑋)
466mrcss 16216 . . . . . . . 8 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 𝑈 𝑈𝑋) → (𝐹𝑥) ⊆ (𝐹 𝑈))
4739, 41, 45, 46syl3anc 1323 . . . . . . 7 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) ∧ 𝑥𝑈) → (𝐹𝑥) ⊆ (𝐹 𝑈))
4847ralrimiva 2962 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → ∀𝑥𝑈 (𝐹𝑥) ⊆ (𝐹 𝑈))
49 sseq1 3611 . . . . . . . 8 (𝑠 = (𝐹𝑥) → (𝑠 ⊆ (𝐹 𝑈) ↔ (𝐹𝑥) ⊆ (𝐹 𝑈)))
5049ralima 6463 . . . . . . 7 ((𝐹 Fn 𝒫 𝑋𝑈 ⊆ 𝒫 𝑋) → (∀𝑠 ∈ (𝐹𝑈)𝑠 ⊆ (𝐹 𝑈) ↔ ∀𝑥𝑈 (𝐹𝑥) ⊆ (𝐹 𝑈)))
5130, 50sylan 488 . . . . . 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 4442 . . . . 5 ( (𝐹𝑈) ⊆ (𝐹 𝑈) ↔ ∀𝑠 ∈ (𝐹𝑈)𝑠 ⊆ (𝐹 𝑈))
5452, 53sylibr 224 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹𝑈) ⊆ (𝐹 𝑈))
556mrcssv 16214 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → (𝐹 𝑈) ⊆ 𝑋)
5655adantr 481 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 𝑈) ⊆ 𝑋)
576mrcss 16216 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ (𝐹𝑈) ⊆ (𝐹 𝑈) ∧ (𝐹 𝑈) ⊆ 𝑋) → (𝐹 (𝐹𝑈)) ⊆ (𝐹‘(𝐹 𝑈)))
581, 54, 56, 57syl3anc 1323 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 (𝐹𝑈)) ⊆ (𝐹‘(𝐹 𝑈)))
596mrcidm 16219 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹‘(𝐹 𝑈)) = (𝐹 𝑈))
601, 44, 59syl2anc 692 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹‘(𝐹 𝑈)) = (𝐹 𝑈))
6158, 60sseqtrd 3626 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 (𝐹𝑈)) ⊆ (𝐹 𝑈))
6238, 61eqssd 3605 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝒫 𝑋) → (𝐹 𝑈) = (𝐹 (𝐹𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  wss 3560  𝒫 cpw 4136   cuni 4409  dom cdm 5084  cima 5087  Fun wfun 5851   Fn wfn 5852  wf 5853  cfv 5857  Moorecmre 16182  mrClscmrc 16183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-int 4448  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-mre 16186  df-mrc 16187
This theorem is referenced by:  mrcun  16222  isacs4lem  17108
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