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Theorem mrcun 17125
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 1138 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 mre1cl 17097 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
3 elpw2g 5237 . . . . . . 7 (𝑋𝐶 → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
42, 3syl 17 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
54biimpar 481 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈 ∈ 𝒫 𝑋)
653adant3 1134 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝑈 ∈ 𝒫 𝑋)
7 elpw2g 5237 . . . . . . 7 (𝑋𝐶 → (𝑉 ∈ 𝒫 𝑋𝑉𝑋))
82, 7syl 17 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝑉 ∈ 𝒫 𝑋𝑉𝑋))
98biimpar 481 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉𝑋) → 𝑉 ∈ 𝒫 𝑋)
1093adant2 1133 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝑉 ∈ 𝒫 𝑋)
116, 10prssd 4735 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → {𝑈, 𝑉} ⊆ 𝒫 𝑋)
12 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
1312mrcuni 17124 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑈, 𝑉} ⊆ 𝒫 𝑋) → (𝐹 {𝑈, 𝑉}) = (𝐹 (𝐹 “ {𝑈, 𝑉})))
141, 11, 13syl2anc 587 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 {𝑈, 𝑉}) = (𝐹 (𝐹 “ {𝑈, 𝑉})))
15 uniprg 4836 . . . 4 ((𝑈 ∈ 𝒫 𝑋𝑉 ∈ 𝒫 𝑋) → {𝑈, 𝑉} = (𝑈𝑉))
166, 10, 15syl2anc 587 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → {𝑈, 𝑉} = (𝑈𝑉))
1716fveq2d 6721 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 {𝑈, 𝑉}) = (𝐹‘(𝑈𝑉)))
1812mrcf 17112 . . . . . . . 8 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
1918ffnd 6546 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
20193ad2ant1 1135 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝐹 Fn 𝒫 𝑋)
21 fnimapr 6795 . . . . . 6 ((𝐹 Fn 𝒫 𝑋𝑈 ∈ 𝒫 𝑋𝑉 ∈ 𝒫 𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹𝑈), (𝐹𝑉)})
2220, 6, 10, 21syl3anc 1373 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹𝑈), (𝐹𝑉)})
2322unieqd 4833 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹𝑈), (𝐹𝑉)})
24 fvex 6730 . . . . 5 (𝐹𝑈) ∈ V
25 fvex 6730 . . . . 5 (𝐹𝑉) ∈ V
2624, 25unipr 4837 . . . 4 {(𝐹𝑈), (𝐹𝑉)} = ((𝐹𝑈) ∪ (𝐹𝑉))
2723, 26eqtrdi 2794 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 “ {𝑈, 𝑉}) = ((𝐹𝑈) ∪ (𝐹𝑉)))
2827fveq2d 6721 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 (𝐹 “ {𝑈, 𝑉})) = (𝐹‘((𝐹𝑈) ∪ (𝐹𝑉))))
2914, 17, 283eqtr3d 2785 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹‘(𝑈𝑉)) = (𝐹‘((𝐹𝑈) ∪ (𝐹𝑉))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1089   = wceq 1543  wcel 2110  cun 3864  wss 3866  𝒫 cpw 4513  {cpr 4543   cuni 4819  cima 5554   Fn wfn 6375  cfv 6380  Moorecmre 17085  mrClscmrc 17086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-mre 17089  df-mrc 17090
This theorem is referenced by: (None)
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