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Theorem mrcun 16329
Description: Idempotence of closure under a pair union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f 𝐹 = (mrCls‘𝐶)
Assertion
Ref Expression
mrcun ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹‘(𝑈𝑉)) = (𝐹‘((𝐹𝑈) ∪ (𝐹𝑉))))

Proof of Theorem mrcun
StepHypRef Expression
1 simp1 1081 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 mre1cl 16301 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
3 elpw2g 4857 . . . . . . 7 (𝑋𝐶 → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
42, 3syl 17 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
54biimpar 501 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈 ∈ 𝒫 𝑋)
653adant3 1101 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝑈 ∈ 𝒫 𝑋)
7 elpw2g 4857 . . . . . . 7 (𝑋𝐶 → (𝑉 ∈ 𝒫 𝑋𝑉𝑋))
82, 7syl 17 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝑉 ∈ 𝒫 𝑋𝑉𝑋))
98biimpar 501 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉𝑋) → 𝑉 ∈ 𝒫 𝑋)
1093adant2 1100 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝑉 ∈ 𝒫 𝑋)
11 prssi 4385 . . . 4 ((𝑈 ∈ 𝒫 𝑋𝑉 ∈ 𝒫 𝑋) → {𝑈, 𝑉} ⊆ 𝒫 𝑋)
126, 10, 11syl2anc 694 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → {𝑈, 𝑉} ⊆ 𝒫 𝑋)
13 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
1413mrcuni 16328 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑈, 𝑉} ⊆ 𝒫 𝑋) → (𝐹 {𝑈, 𝑉}) = (𝐹 (𝐹 “ {𝑈, 𝑉})))
151, 12, 14syl2anc 694 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 {𝑈, 𝑉}) = (𝐹 (𝐹 “ {𝑈, 𝑉})))
16 uniprg 4482 . . . 4 ((𝑈 ∈ 𝒫 𝑋𝑉 ∈ 𝒫 𝑋) → {𝑈, 𝑉} = (𝑈𝑉))
176, 10, 16syl2anc 694 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → {𝑈, 𝑉} = (𝑈𝑉))
1817fveq2d 6233 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 {𝑈, 𝑉}) = (𝐹‘(𝑈𝑉)))
1913mrcf 16316 . . . . . . . 8 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
20 ffn 6083 . . . . . . . 8 (𝐹:𝒫 𝑋𝐶𝐹 Fn 𝒫 𝑋)
2119, 20syl 17 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
22213ad2ant1 1102 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝐹 Fn 𝒫 𝑋)
23 fnimapr 6301 . . . . . 6 ((𝐹 Fn 𝒫 𝑋𝑈 ∈ 𝒫 𝑋𝑉 ∈ 𝒫 𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹𝑈), (𝐹𝑉)})
2422, 6, 10, 23syl3anc 1366 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹𝑈), (𝐹𝑉)})
2524unieqd 4478 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹𝑈), (𝐹𝑉)})
26 fvex 6239 . . . . 5 (𝐹𝑈) ∈ V
27 fvex 6239 . . . . 5 (𝐹𝑉) ∈ V
2826, 27unipr 4481 . . . 4 {(𝐹𝑈), (𝐹𝑉)} = ((𝐹𝑈) ∪ (𝐹𝑉))
2925, 28syl6eq 2701 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 “ {𝑈, 𝑉}) = ((𝐹𝑈) ∪ (𝐹𝑉)))
3029fveq2d 6233 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 (𝐹 “ {𝑈, 𝑉})) = (𝐹‘((𝐹𝑈) ∪ (𝐹𝑉))))
3115, 18, 303eqtr3d 2693 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹‘(𝑈𝑉)) = (𝐹‘((𝐹𝑈) ∪ (𝐹𝑉))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1054   = wceq 1523  wcel 2030  cun 3605  wss 3607  𝒫 cpw 4191  {cpr 4212   cuni 4468  cima 5146   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: (None)
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