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Theorem mrcss 17596
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 3987 . . . . . 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 4071 . . . 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 1132 . 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 1134 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 βŠ† 𝑋) β†’ 𝐢 ∈ (Mooreβ€˜π‘‹))
8 sstr 3988 . . . 4 ((π‘ˆ βŠ† 𝑉 ∧ 𝑉 βŠ† 𝑋) β†’ π‘ˆ βŠ† 𝑋)
983adant1 1128 . . 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 17590 . . 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 17590 . . 3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑉 βŠ† 𝑋) β†’ (πΉβ€˜π‘‰) = ∩ {𝑠 ∈ 𝐢 ∣ 𝑉 βŠ† 𝑠})
14133adant2 1129 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑉 βŠ† 𝑋) β†’ (πΉβ€˜π‘‰) = ∩ {𝑠 ∈ 𝐢 ∣ 𝑉 βŠ† 𝑠})
156, 12, 143sstr4d 4027 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 1085   = wceq 1534   ∈ wcel 2099  {crab 3429   βŠ† wss 3947  βˆ© cint 4949  β€˜cfv 6548  Moorecmre 17562  mrClscmrc 17563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-mre 17566  df-mrc 17567
This theorem is referenced by:  mrcsscl  17600  mrcuni  17601  mrcssd  17604  ismrc  42121  isnacs3  42130
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