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Theorem fnmrc 16468
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 16448 . . 3 mrCls = (𝑐 ran Moore ↦ (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}))
21fnmpt 6227 . 2 (∀𝑐 ran Moore(𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V → mrCls Fn ran Moore)
3 mreunirn 16462 . . 3 (𝑐 ran Moore ↔ 𝑐 ∈ (Moore‘ 𝑐))
4 mrcflem 16467 . . . . 5 (𝑐 ∈ (Moore‘ 𝑐) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}):𝒫 𝑐𝑐)
5 fssxp 6271 . . . . 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 7180 . . . . . 6 𝑐 ∈ V
87pwex 5050 . . . . 5 𝒫 𝑐 ∈ V
9 vex 3394 . . . . 5 𝑐 ∈ V
108, 9xpex 7188 . . . 4 (𝒫 𝑐 × 𝑐) ∈ V
11 ssexg 4999 . . . 4 (((𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ⊆ (𝒫 𝑐 × 𝑐) ∧ (𝒫 𝑐 × 𝑐) ∈ V) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
126, 10, 11sylancl 576 . . 3 (𝑐 ∈ (Moore‘ 𝑐) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
133, 12sylbi 208 . 2 (𝑐 ran Moore → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
142, 13mprg 3114 1 mrCls Fn ran Moore
Colors of variables: wff setvar class
Syntax hints:  wcel 2156  {crab 3100  Vcvv 3391  wss 3769  𝒫 cpw 4351   cuni 4630   cint 4669  cmpt 4923   × cxp 5309  ran crn 5312   Fn wfn 6092  wf 6093  cfv 6097  Moorecmre 16443  mrClscmrc 16444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-int 4670  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-fv 6105  df-mre 16447  df-mrc 16448
This theorem is referenced by:  ismrc  37760
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