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Theorem mrcsscl 17426
Description: The closure is the minimal closed set; any closed set which contains the generators is a superset of the closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f 𝐹 = (mrClsβ€˜πΆ)
Assertion
Ref Expression
mrcsscl ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 ∈ 𝐢) β†’ (πΉβ€˜π‘ˆ) βŠ† 𝑉)

Proof of Theorem mrcsscl
StepHypRef Expression
1 mress 17399 . . . 4 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑉 ∈ 𝐢) β†’ 𝑉 βŠ† 𝑋)
213adant2 1130 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 ∈ 𝐢) β†’ 𝑉 βŠ† 𝑋)
3 mrcfval.f . . . 4 𝐹 = (mrClsβ€˜πΆ)
43mrcss 17422 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 βŠ† 𝑋) β†’ (πΉβ€˜π‘ˆ) βŠ† (πΉβ€˜π‘‰))
52, 4syld3an3 1408 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 ∈ 𝐢) β†’ (πΉβ€˜π‘ˆ) βŠ† (πΉβ€˜π‘‰))
63mrcid 17419 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑉 ∈ 𝐢) β†’ (πΉβ€˜π‘‰) = 𝑉)
763adant2 1130 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 ∈ 𝐢) β†’ (πΉβ€˜π‘‰) = 𝑉)
85, 7sseqtrd 3972 1 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 ∈ 𝐢) β†’ (πΉβ€˜π‘ˆ) βŠ† 𝑉)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   βŠ† wss 3898  β€˜cfv 6479  Moorecmre 17388  mrClscmrc 17389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-fv 6487  df-mre 17392  df-mrc 17393
This theorem is referenced by:  submrc  17434  isacs2  17459  isacs3lem  18357  mrelatlub  18377  mndind  18563  gsumwspan  18581  symggen  19174  cntzspan  19540  dprdspan  19725  subgdmdprd  19732  subgdprd  19733  dprdsn  19734  dprd2dlem1  19739  dprd2da  19740  dmdprdsplit2lem  19743  ablfac1b  19768  pgpfac1lem1  19772  pgpfac1lem5  19777  mrccss  21005  evlseu  21399  ismrcd2  40783  mrefg3  40792  isnacs3  40794
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