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Theorem mrcss 17556
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 3988 . . . . . 6 (π‘ˆ βŠ† 𝑉 β†’ (𝑉 βŠ† 𝑠 β†’ π‘ˆ βŠ† 𝑠))
21adantr 481 . . . . 5 ((π‘ˆ βŠ† 𝑉 ∧ 𝑠 ∈ 𝐢) β†’ (𝑉 βŠ† 𝑠 β†’ π‘ˆ βŠ† 𝑠))
32ss2rabdv 4072 . . . 4 (π‘ˆ βŠ† 𝑉 β†’ {𝑠 ∈ 𝐢 ∣ 𝑉 βŠ† 𝑠} βŠ† {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
4 intss 4972 . . . 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 1134 . 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 1136 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 βŠ† 𝑋) β†’ 𝐢 ∈ (Mooreβ€˜π‘‹))
8 sstr 3989 . . . 4 ((π‘ˆ βŠ† 𝑉 ∧ 𝑉 βŠ† 𝑋) β†’ π‘ˆ βŠ† 𝑋)
983adant1 1130 . . 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 17550 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑋) β†’ (πΉβ€˜π‘ˆ) = ∩ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
127, 9, 11syl2anc 584 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 βŠ† 𝑋) β†’ (πΉβ€˜π‘ˆ) = ∩ {𝑠 ∈ 𝐢 ∣ π‘ˆ βŠ† 𝑠})
1310mrcval 17550 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑉 βŠ† 𝑋) β†’ (πΉβ€˜π‘‰) = ∩ {𝑠 ∈ 𝐢 ∣ 𝑉 βŠ† 𝑠})
14133adant2 1131 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 βŠ† 𝑋) β†’ (πΉβ€˜π‘‰) = ∩ {𝑠 ∈ 𝐢 ∣ 𝑉 βŠ† 𝑠})
156, 12, 143sstr4d 4028 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 1087   = wceq 1541   ∈ wcel 2106  {crab 3432   βŠ† wss 3947  βˆ© cint 4949  β€˜cfv 6540  Moorecmre 17522  mrClscmrc 17523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-mre 17526  df-mrc 17527
This theorem is referenced by:  mrcsscl  17560  mrcuni  17561  mrcssd  17564  ismrc  41424  isnacs3  41433
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