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Theorem mrcun 17666
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 1135 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 mre1cl 17638 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
3 elpw2g 5338 . . . . . . 7 (𝑋𝐶 → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
42, 3syl 17 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
54biimpar 477 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈 ∈ 𝒫 𝑋)
653adant3 1131 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝑈 ∈ 𝒫 𝑋)
7 elpw2g 5338 . . . . . . 7 (𝑋𝐶 → (𝑉 ∈ 𝒫 𝑋𝑉𝑋))
82, 7syl 17 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝑉 ∈ 𝒫 𝑋𝑉𝑋))
98biimpar 477 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉𝑋) → 𝑉 ∈ 𝒫 𝑋)
1093adant2 1130 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝑉 ∈ 𝒫 𝑋)
116, 10prssd 4826 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → {𝑈, 𝑉} ⊆ 𝒫 𝑋)
12 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
1312mrcuni 17665 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑈, 𝑉} ⊆ 𝒫 𝑋) → (𝐹 {𝑈, 𝑉}) = (𝐹 (𝐹 “ {𝑈, 𝑉})))
141, 11, 13syl2anc 584 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 {𝑈, 𝑉}) = (𝐹 (𝐹 “ {𝑈, 𝑉})))
15 uniprg 4927 . . . 4 ((𝑈 ∈ 𝒫 𝑋𝑉 ∈ 𝒫 𝑋) → {𝑈, 𝑉} = (𝑈𝑉))
166, 10, 15syl2anc 584 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → {𝑈, 𝑉} = (𝑈𝑉))
1716fveq2d 6910 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 {𝑈, 𝑉}) = (𝐹‘(𝑈𝑉)))
1812mrcf 17653 . . . . . . . 8 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
1918ffnd 6737 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
20193ad2ant1 1132 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝐹 Fn 𝒫 𝑋)
21 fnimapr 6991 . . . . . 6 ((𝐹 Fn 𝒫 𝑋𝑈 ∈ 𝒫 𝑋𝑉 ∈ 𝒫 𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹𝑈), (𝐹𝑉)})
2220, 6, 10, 21syl3anc 1370 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹𝑈), (𝐹𝑉)})
2322unieqd 4924 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹𝑈), (𝐹𝑉)})
24 fvex 6919 . . . . 5 (𝐹𝑈) ∈ V
25 fvex 6919 . . . . 5 (𝐹𝑉) ∈ V
2624, 25unipr 4928 . . . 4 {(𝐹𝑈), (𝐹𝑉)} = ((𝐹𝑈) ∪ (𝐹𝑉))
2723, 26eqtrdi 2790 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 “ {𝑈, 𝑉}) = ((𝐹𝑈) ∪ (𝐹𝑉)))
2827fveq2d 6910 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 (𝐹 “ {𝑈, 𝑉})) = (𝐹‘((𝐹𝑈) ∪ (𝐹𝑉))))
2914, 17, 283eqtr3d 2782 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹‘(𝑈𝑉)) = (𝐹‘((𝐹𝑈) ∪ (𝐹𝑉))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1536  wcel 2105  cun 3960  wss 3962  𝒫 cpw 4604  {cpr 4632   cuni 4911  cima 5691   Fn wfn 6557  cfv 6562  Moorecmre 17626  mrClscmrc 17627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-mre 17630  df-mrc 17631
This theorem is referenced by: (None)
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