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Theorem mrcss 17565
Description: Closure preserves subset ordering. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f 𝐹 = (mrClsβ€˜πΆ)
Assertion
Ref Expression
mrcss ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 βŠ† 𝑋) β†’ (πΉβ€˜π‘ˆ) βŠ† (πΉβ€˜π‘‰))
Proof of Theorem mrcss
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 sstr2 3982 . . . . . 6 (π‘ˆ βŠ† 𝑉 β†’ (𝑉 βŠ† 𝑠 β†’ π‘ˆ βŠ† 𝑠))
21adantr 480 . . . . 5 ((π‘ˆ βŠ† 𝑉 ∧ 𝑠 ∈ 𝐢) β†’ (𝑉 βŠ† 𝑠 β†’ π‘ˆ βŠ† 𝑠))
32ss2rabdv 4066 . . . 4 (π‘ˆ βŠ† 𝑉 β†’ {𝑠 ∈ 𝐢 ∣ 𝑉 βŠ† 𝑠} βŠ† {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
4 intss 4964 . . . 4 ({𝑠 ∈ 𝐢 ∣ 𝑉 βŠ† 𝑠} βŠ† {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠} β†’ ∩ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠} βŠ† ∩ {𝑠 ∈ 𝐢 ∣ 𝑉 βŠ† 𝑠})
53, 4syl 17 . . 3 (π‘ˆ βŠ† 𝑉 β†’ ∩ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠} βŠ† ∩ {𝑠 ∈ 𝐢 ∣ 𝑉 βŠ† 𝑠})
653ad2ant2 1131 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 βŠ† 𝑋) β†’ ∩ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠} βŠ† ∩ {𝑠 ∈ 𝐢 ∣ 𝑉 βŠ† 𝑠})
7 simp1 1133 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 βŠ† 𝑋) β†’ 𝐢 ∈ (Mooreβ€˜π‘‹))
8 sstr 3983 . . . 4 ((π‘ˆ βŠ† 𝑉 ∧ 𝑉 βŠ† 𝑋) β†’ π‘ˆ βŠ† 𝑋)
983adant1 1127 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 βŠ† 𝑋) β†’ π‘ˆ βŠ† 𝑋)
10 mrcfval.f . . . 4 𝐹 = (mrClsβ€˜πΆ)
1110mrcval 17559 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ (πΉβ€˜π‘ˆ) = ∩ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
127, 9, 11syl2anc 583 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 βŠ† 𝑋) β†’ (πΉβ€˜π‘ˆ) = ∩ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
1310mrcval 17559 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑉 βŠ† 𝑋) β†’ (πΉβ€˜π‘‰) = ∩ {𝑠 ∈ 𝐢 ∣ 𝑉 βŠ† 𝑠})
14133adant2 1128 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 βŠ† 𝑋) β†’ (πΉβ€˜π‘‰) = ∩ {𝑠 ∈ 𝐢 ∣ 𝑉 βŠ† 𝑠})
156, 12, 143sstr4d 4022 1 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 βŠ† 𝑋) β†’ (πΉβ€˜π‘ˆ) βŠ† (πΉβ€˜π‘‰))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {crab 3424   βŠ† wss 3941  βˆ© cint 4941  β€˜cfv 6534  Moorecmre 17531  mrClscmrc 17532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-mre 17535  df-mrc 17536
This theorem is referenced by:  mrcsscl  17569  mrcuni  17570  mrcssd  17573  ismrc  41991  isnacs3  42000
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