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Theorem fnmrc 16869
Description: Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fnmrc mrCls Fn ran Moore

Proof of Theorem fnmrc
Dummy variables 𝑐 𝑥 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mrc 16849 . . 3 mrCls = (𝑐 ran Moore ↦ (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}))
21fnmpt 6468 . 2 (∀𝑐 ran Moore(𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V → mrCls Fn ran Moore)
3 mreunirn 16863 . . 3 (𝑐 ran Moore ↔ 𝑐 ∈ (Moore‘ 𝑐))
4 mrcflem 16868 . . . . 5 (𝑐 ∈ (Moore‘ 𝑐) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}):𝒫 𝑐𝑐)
5 fssxp 6515 . . . . 5 ((𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}):𝒫 𝑐𝑐 → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ⊆ (𝒫 𝑐 × 𝑐))
64, 5syl 17 . . . 4 (𝑐 ∈ (Moore‘ 𝑐) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ⊆ (𝒫 𝑐 × 𝑐))
7 vuniex 7450 . . . . . 6 𝑐 ∈ V
87pwex 5258 . . . . 5 𝒫 𝑐 ∈ V
9 vex 3472 . . . . 5 𝑐 ∈ V
108, 9xpex 7461 . . . 4 (𝒫 𝑐 × 𝑐) ∈ V
11 ssexg 5203 . . . 4 (((𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ⊆ (𝒫 𝑐 × 𝑐) ∧ (𝒫 𝑐 × 𝑐) ∈ V) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
126, 10, 11sylancl 589 . . 3 (𝑐 ∈ (Moore‘ 𝑐) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
133, 12sylbi 220 . 2 (𝑐 ran Moore → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
142, 13mprg 3144 1 mrCls Fn ran Moore
Colors of variables: wff setvar class
Syntax hints:  wcel 2114  {crab 3134  Vcvv 3469  wss 3908  𝒫 cpw 4511   cuni 4813   cint 4851  cmpt 5122   × cxp 5530  ran crn 5533   Fn wfn 6329  wf 6330  cfv 6334  Moorecmre 16844  mrClscmrc 16845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-int 4852  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-mre 16848  df-mrc 16849
This theorem is referenced by:  ismrc  39572
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