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Theorem fnmrc 16183
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 16163 . . 3 mrCls = (𝑐 ran Moore ↦ (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}))
21fnmpt 5979 . 2 (∀𝑐 ran Moore(𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V → mrCls Fn ran Moore)
3 mreunirn 16177 . . 3 (𝑐 ran Moore ↔ 𝑐 ∈ (Moore‘ 𝑐))
4 mrcflem 16182 . . . . 5 (𝑐 ∈ (Moore‘ 𝑐) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}):𝒫 𝑐𝑐)
5 fssxp 6019 . . . . 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 6908 . . . . . 6 𝑐 ∈ V
87pwex 4813 . . . . 5 𝒫 𝑐 ∈ V
9 vex 3194 . . . . 5 𝑐 ∈ V
108, 9xpex 6916 . . . 4 (𝒫 𝑐 × 𝑐) ∈ V
11 ssexg 4769 . . . 4 (((𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ⊆ (𝒫 𝑐 × 𝑐) ∧ (𝒫 𝑐 × 𝑐) ∈ V) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
126, 10, 11sylancl 693 . . 3 (𝑐 ∈ (Moore‘ 𝑐) → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
133, 12sylbi 207 . 2 (𝑐 ran Moore → (𝑥 ∈ 𝒫 𝑐 {𝑠𝑐𝑥𝑠}) ∈ V)
142, 13mprg 2926 1 mrCls Fn ran Moore
Colors of variables: wff setvar class
Syntax hints:  wcel 1992  {crab 2916  Vcvv 3191  wss 3560  𝒫 cpw 4135   cuni 4407   cint 4445  cmpt 4678   × cxp 5077  ran crn 5080   Fn wfn 5845  wf 5846  cfv 5850  Moorecmre 16158  mrClscmrc 16159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-mre 16162  df-mrc 16163
This theorem is referenced by:  ismrc  36730
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