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Theorem mrcfval 16475
Description: Value of the function expression for the Moore closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f 𝐹 = (mrCls‘𝐶)
Assertion
Ref Expression
mrcfval (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
Distinct variable groups:   𝑥,𝐹,𝑠   𝑥,𝐶,𝑠   𝑥,𝑋,𝑠

Proof of Theorem mrcfval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 mrcfval.f . 2 𝐹 = (mrCls‘𝐶)
2 fvssunirn 6358 . . . . 5 (Moore‘𝑋) ⊆ ran Moore
32sseli 3748 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝐶 ran Moore)
4 unieq 4582 . . . . . . 7 (𝑐 = 𝐶 𝑐 = 𝐶)
54pweqd 4302 . . . . . 6 (𝑐 = 𝐶 → 𝒫 𝑐 = 𝒫 𝐶)
6 rabeq 3342 . . . . . . 7 (𝑐 = 𝐶 → {𝑠𝑐𝑥𝑠} = {𝑠𝐶𝑥𝑠})
76inteqd 4616 . . . . . 6 (𝑐 = 𝐶 {𝑠𝑐𝑥𝑠} = {𝑠𝐶𝑥𝑠})
85, 7mpteq12dv 4867 . . . . 5 (𝑐 = 𝐶 → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) = (𝑥 ∈ 𝒫 𝐶 {𝑠𝐶𝑥𝑠}))
9 df-mrc 16454 . . . . 5 mrCls = (𝑐 ran Moore ↦ (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}))
10 mreunirn 16468 . . . . . . . 8 (𝑐 ran Moore ↔ 𝑐 ∈ (Moore‘ 𝑐))
11 mrcflem 16473 . . . . . . . 8 (𝑐 ∈ (Moore‘ 𝑐) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}):𝒫 𝑐𝑐)
1210, 11sylbi 207 . . . . . . 7 (𝑐 ran Moore → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}):𝒫 𝑐𝑐)
13 fssxp 6200 . . . . . . 7 ((𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}):𝒫 𝑐𝑐 → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ⊆ (𝒫 𝑐 × 𝑐))
1412, 13syl 17 . . . . . 6 (𝑐 ran Moore → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ⊆ (𝒫 𝑐 × 𝑐))
15 vuniex 7100 . . . . . . . 8 𝑐 ∈ V
1615pwex 4979 . . . . . . 7 𝒫 𝑐 ∈ V
17 vex 3354 . . . . . . 7 𝑐 ∈ V
1816, 17xpex 7108 . . . . . 6 (𝒫 𝑐 × 𝑐) ∈ V
19 ssexg 4938 . . . . . 6 (((𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ⊆ (𝒫 𝑐 × 𝑐) ∧ (𝒫 𝑐 × 𝑐) ∈ V) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
2014, 18, 19sylancl 566 . . . . 5 (𝑐 ran Moore → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
218, 9, 20fvmpt3 6428 . . . 4 (𝐶 ran Moore → (mrCls‘𝐶) = (𝑥 ∈ 𝒫 𝐶 {𝑠𝐶𝑥𝑠}))
223, 21syl 17 . . 3 (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) = (𝑥 ∈ 𝒫 𝐶 {𝑠𝐶𝑥𝑠}))
23 mreuni 16467 . . . . 5 (𝐶 ∈ (Moore‘𝑋) → 𝐶 = 𝑋)
2423pweqd 4302 . . . 4 (𝐶 ∈ (Moore‘𝑋) → 𝒫 𝐶 = 𝒫 𝑋)
2524mpteq1d 4872 . . 3 (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝐶 {𝑠𝐶𝑥𝑠}) = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
2622, 25eqtrd 2805 . 2 (𝐶 ∈ (Moore‘𝑋) → (mrCls‘𝐶) = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
271, 26syl5eq 2817 1 (𝐶 ∈ (Moore‘𝑋) → 𝐹 = (𝑥 ∈ 𝒫 𝑋 {𝑠𝐶𝑥𝑠}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351  wss 3723  𝒫 cpw 4297   cuni 4574   cint 4611  cmpt 4863   × cxp 5247  ran crn 5250  wf 6027  cfv 6031  Moorecmre 16449  mrClscmrc 16450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-mre 16453  df-mrc 16454
This theorem is referenced by:  mrcf  16476  mrcval  16477  acsficl2d  17383  mrclsp  19201  mrccls  21103
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