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Theorem fnmrc 17665
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 17645 . . 3 mrCls = (𝑐 ran Moore ↦ (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}))
21fnmpt 6720 . 2 (∀𝑐 ran Moore(𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V → mrCls Fn ran Moore)
3 mreunirn 17659 . . 3 (𝑐 ran Moore ↔ 𝑐 ∈ (Moore‘ 𝑐))
4 mrcflem 17664 . . . . 5 (𝑐 ∈ (Moore‘ 𝑐) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}):𝒫 𝑐𝑐)
5 fssxp 6775 . . . . 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 7774 . . . . . 6 𝑐 ∈ V
87pwex 5398 . . . . 5 𝒫 𝑐 ∈ V
9 vex 3492 . . . . 5 𝑐 ∈ V
108, 9xpex 7788 . . . 4 (𝒫 𝑐 × 𝑐) ∈ V
11 ssexg 5341 . . . 4 (((𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ⊆ (𝒫 𝑐 × 𝑐) ∧ (𝒫 𝑐 × 𝑐) ∈ V) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
126, 10, 11sylancl 585 . . 3 (𝑐 ∈ (Moore‘ 𝑐) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
133, 12sylbi 217 . 2 (𝑐 ran Moore → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
142, 13mprg 3073 1 mrCls Fn ran Moore
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  {crab 3443  Vcvv 3488  wss 3976  𝒫 cpw 4622   cuni 4931   cint 4970  cmpt 5249   × cxp 5698  ran crn 5701   Fn wfn 6568  wf 6569  cfv 6573  Moorecmre 17640  mrClscmrc 17641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-mre 17644  df-mrc 17645
This theorem is referenced by:  ismrc  42657
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