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Theorem mrcun 16105
 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 1054 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 mre1cl 16077 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
3 elpw2g 4754 . . . . . . 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 1074 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝑈 ∈ 𝒫 𝑋)
7 elpw2g 4754 . . . . . . 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 1073 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝑉 ∈ 𝒫 𝑋)
11 prssi 4293 . . . 4 ((𝑈 ∈ 𝒫 𝑋𝑉 ∈ 𝒫 𝑋) → {𝑈, 𝑉} ⊆ 𝒫 𝑋)
126, 10, 11syl2anc 691 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → {𝑈, 𝑉} ⊆ 𝒫 𝑋)
13 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
1413mrcuni 16104 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑈, 𝑉} ⊆ 𝒫 𝑋) → (𝐹 {𝑈, 𝑉}) = (𝐹 (𝐹 “ {𝑈, 𝑉})))
151, 12, 14syl2anc 691 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 {𝑈, 𝑉}) = (𝐹 (𝐹 “ {𝑈, 𝑉})))
16 uniprg 4386 . . . 4 ((𝑈 ∈ 𝒫 𝑋𝑉 ∈ 𝒫 𝑋) → {𝑈, 𝑉} = (𝑈𝑉))
176, 10, 16syl2anc 691 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → {𝑈, 𝑉} = (𝑈𝑉))
1817fveq2d 6107 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 {𝑈, 𝑉}) = (𝐹‘(𝑈𝑉)))
1913mrcf 16092 . . . . . . . 8 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
20 ffn 5958 . . . . . . . 8 (𝐹:𝒫 𝑋𝐶𝐹 Fn 𝒫 𝑋)
2119, 20syl 17 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
22213ad2ant1 1075 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝐹 Fn 𝒫 𝑋)
23 fnimapr 6172 . . . . . 6 ((𝐹 Fn 𝒫 𝑋𝑈 ∈ 𝒫 𝑋𝑉 ∈ 𝒫 𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹𝑈), (𝐹𝑉)})
2422, 6, 10, 23syl3anc 1318 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹𝑈), (𝐹𝑉)})
2524unieqd 4382 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹𝑈), (𝐹𝑉)})
26 fvex 6113 . . . . 5 (𝐹𝑈) ∈ V
27 fvex 6113 . . . . 5 (𝐹𝑉) ∈ V
2826, 27unipr 4385 . . . 4 {(𝐹𝑈), (𝐹𝑉)} = ((𝐹𝑈) ∪ (𝐹𝑉))
2925, 28syl6eq 2660 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 “ {𝑈, 𝑉}) = ((𝐹𝑈) ∪ (𝐹𝑉)))
3029fveq2d 6107 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 (𝐹 “ {𝑈, 𝑉})) = (𝐹‘((𝐹𝑈) ∪ (𝐹𝑉))))
3115, 18, 303eqtr3d 2652 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹‘(𝑈𝑉)) = (𝐹‘((𝐹𝑈) ∪ (𝐹𝑉))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540  𝒫 cpw 4108  {cpr 4127  ∪ cuni 4372   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  Moorecmre 16065  mrClscmrc 16066 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-mre 16069  df-mrc 16070 This theorem is referenced by: (None)
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