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Theorem fnmrc 16358
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 16338 . . 3 mrCls = (𝑐 ran Moore ↦ (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}))
21fnmpt 6101 . 2 (∀𝑐 ran Moore(𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V → mrCls Fn ran Moore)
3 mreunirn 16352 . . 3 (𝑐 ran Moore ↔ 𝑐 ∈ (Moore‘ 𝑐))
4 mrcflem 16357 . . . . 5 (𝑐 ∈ (Moore‘ 𝑐) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}):𝒫 𝑐𝑐)
5 fssxp 6141 . . . . 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 7039 . . . . . 6 𝑐 ∈ V
87pwex 4921 . . . . 5 𝒫 𝑐 ∈ V
9 vex 3275 . . . . 5 𝑐 ∈ V
108, 9xpex 7047 . . . 4 (𝒫 𝑐 × 𝑐) ∈ V
11 ssexg 4880 . . . 4 (((𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ⊆ (𝒫 𝑐 × 𝑐) ∧ (𝒫 𝑐 × 𝑐) ∈ V) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
126, 10, 11sylancl 697 . . 3 (𝑐 ∈ (Moore‘ 𝑐) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
133, 12sylbi 207 . 2 (𝑐 ran Moore → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
142, 13mprg 2996 1 mrCls Fn ran Moore
Colors of variables: wff setvar class
Syntax hints:  wcel 2071  {crab 2986  Vcvv 3272  wss 3648  𝒫 cpw 4234   cuni 4512   cint 4551  cmpt 4805   × cxp 5184  ran crn 5187   Fn wfn 5964  wf 5965  cfv 5969  Moorecmre 16333  mrClscmrc 16334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-8 2073  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-sep 4857  ax-nul 4865  ax-pow 4916  ax-pr 4979  ax-un 7034
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-rex 2988  df-rab 2991  df-v 3274  df-sbc 3510  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-nul 3992  df-if 4163  df-pw 4236  df-sn 4254  df-pr 4256  df-op 4260  df-uni 4513  df-int 4552  df-br 4729  df-opab 4789  df-mpt 4806  df-id 5096  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-rn 5197  df-res 5198  df-ima 5199  df-iota 5932  df-fun 5971  df-fn 5972  df-f 5973  df-fv 5977  df-mre 16337  df-mrc 16338
This theorem is referenced by:  ismrc  37651
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