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Theorem fnmrc 17422
Description: Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fnmrc mrCls Fn βˆͺ ran Moore

Proof of Theorem fnmrc
Dummy variables 𝑐 π‘₯ 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mrc 17402 . . 3 mrCls = (𝑐 ∈ βˆͺ ran Moore ↦ (π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}))
21fnmpt 6637 . 2 (βˆ€π‘ ∈ βˆͺ ran Moore(π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}) ∈ V β†’ mrCls Fn βˆͺ ran Moore)
3 mreunirn 17416 . . 3 (𝑐 ∈ βˆͺ ran Moore ↔ 𝑐 ∈ (Mooreβ€˜βˆͺ 𝑐))
4 mrcflem 17421 . . . . 5 (𝑐 ∈ (Mooreβ€˜βˆͺ 𝑐) β†’ (π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}):𝒫 βˆͺ π‘βŸΆπ‘)
5 fssxp 6692 . . . . 5 ((π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}):𝒫 βˆͺ π‘βŸΆπ‘ β†’ (π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}) βŠ† (𝒫 βˆͺ 𝑐 Γ— 𝑐))
64, 5syl 17 . . . 4 (𝑐 ∈ (Mooreβ€˜βˆͺ 𝑐) β†’ (π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}) βŠ† (𝒫 βˆͺ 𝑐 Γ— 𝑐))
7 vuniex 7667 . . . . . 6 βˆͺ 𝑐 ∈ V
87pwex 5334 . . . . 5 𝒫 βˆͺ 𝑐 ∈ V
9 vex 3448 . . . . 5 𝑐 ∈ V
108, 9xpex 7678 . . . 4 (𝒫 βˆͺ 𝑐 Γ— 𝑐) ∈ V
11 ssexg 5279 . . . 4 (((π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}) βŠ† (𝒫 βˆͺ 𝑐 Γ— 𝑐) ∧ (𝒫 βˆͺ 𝑐 Γ— 𝑐) ∈ V) β†’ (π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}) ∈ V)
126, 10, 11sylancl 587 . . 3 (𝑐 ∈ (Mooreβ€˜βˆͺ 𝑐) β†’ (π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}) ∈ V)
133, 12sylbi 216 . 2 (𝑐 ∈ βˆͺ ran Moore β†’ (π‘₯ ∈ 𝒫 βˆͺ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ π‘₯ βŠ† 𝑠}) ∈ V)
142, 13mprg 3069 1 mrCls Fn βˆͺ ran Moore
Colors of variables: wff setvar class
Syntax hints:   ∈ wcel 2107  {crab 3406  Vcvv 3444   βŠ† wss 3909  π’« cpw 4559  βˆͺ cuni 4864  βˆ© cint 4906   ↦ cmpt 5187   Γ— cxp 5629  ran crn 5632   Fn wfn 6487  βŸΆwf 6488  β€˜cfv 6492  Moorecmre 17397  mrClscmrc 17398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-mre 17401  df-mrc 17402
This theorem is referenced by:  ismrc  40890
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