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Theorem mrcun 16276
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 1060 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 mre1cl 16248 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → 𝑋𝐶)
3 elpw2g 4825 . . . . . . 7 (𝑋𝐶 → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
42, 3syl 17 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝑈 ∈ 𝒫 𝑋𝑈𝑋))
54biimpar 502 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → 𝑈 ∈ 𝒫 𝑋)
653adant3 1080 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝑈 ∈ 𝒫 𝑋)
7 elpw2g 4825 . . . . . . 7 (𝑋𝐶 → (𝑉 ∈ 𝒫 𝑋𝑉𝑋))
82, 7syl 17 . . . . . 6 (𝐶 ∈ (Moore‘𝑋) → (𝑉 ∈ 𝒫 𝑋𝑉𝑋))
98biimpar 502 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉𝑋) → 𝑉 ∈ 𝒫 𝑋)
1093adant2 1079 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝑉 ∈ 𝒫 𝑋)
11 prssi 4351 . . . 4 ((𝑈 ∈ 𝒫 𝑋𝑉 ∈ 𝒫 𝑋) → {𝑈, 𝑉} ⊆ 𝒫 𝑋)
126, 10, 11syl2anc 693 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → {𝑈, 𝑉} ⊆ 𝒫 𝑋)
13 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
1413mrcuni 16275 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑈, 𝑉} ⊆ 𝒫 𝑋) → (𝐹 {𝑈, 𝑉}) = (𝐹 (𝐹 “ {𝑈, 𝑉})))
151, 12, 14syl2anc 693 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 {𝑈, 𝑉}) = (𝐹 (𝐹 “ {𝑈, 𝑉})))
16 uniprg 4448 . . . 4 ((𝑈 ∈ 𝒫 𝑋𝑉 ∈ 𝒫 𝑋) → {𝑈, 𝑉} = (𝑈𝑉))
176, 10, 16syl2anc 693 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → {𝑈, 𝑉} = (𝑈𝑉))
1817fveq2d 6193 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 {𝑈, 𝑉}) = (𝐹‘(𝑈𝑉)))
1913mrcf 16263 . . . . . . . 8 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
20 ffn 6043 . . . . . . . 8 (𝐹:𝒫 𝑋𝐶𝐹 Fn 𝒫 𝑋)
2119, 20syl 17 . . . . . . 7 (𝐶 ∈ (Moore‘𝑋) → 𝐹 Fn 𝒫 𝑋)
22213ad2ant1 1081 . . . . . 6 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → 𝐹 Fn 𝒫 𝑋)
23 fnimapr 6260 . . . . . 6 ((𝐹 Fn 𝒫 𝑋𝑈 ∈ 𝒫 𝑋𝑉 ∈ 𝒫 𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹𝑈), (𝐹𝑉)})
2422, 6, 10, 23syl3anc 1325 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹𝑈), (𝐹𝑉)})
2524unieqd 4444 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 “ {𝑈, 𝑉}) = {(𝐹𝑈), (𝐹𝑉)})
26 fvex 6199 . . . . 5 (𝐹𝑈) ∈ V
27 fvex 6199 . . . . 5 (𝐹𝑉) ∈ V
2826, 27unipr 4447 . . . 4 {(𝐹𝑈), (𝐹𝑉)} = ((𝐹𝑈) ∪ (𝐹𝑉))
2925, 28syl6eq 2671 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 “ {𝑈, 𝑉}) = ((𝐹𝑈) ∪ (𝐹𝑉)))
3029fveq2d 6193 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹 (𝐹 “ {𝑈, 𝑉})) = (𝐹‘((𝐹𝑈) ∪ (𝐹𝑉))))
3115, 18, 303eqtr3d 2663 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋𝑉𝑋) → (𝐹‘(𝑈𝑉)) = (𝐹‘((𝐹𝑈) ∪ (𝐹𝑉))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1037   = wceq 1482  wcel 1989  cun 3570  wss 3572  𝒫 cpw 4156  {cpr 4177   cuni 4434  cima 5115   Fn wfn 5881  wf 5882  cfv 5886  Moorecmre 16236  mrClscmrc 16237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-int 4474  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-mre 16240  df-mrc 16241
This theorem is referenced by: (None)
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