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Theorem mrcss 17660
Description: Closure preserves subset ordering. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f 𝐹 = (mrCls‘𝐶)
Assertion
Ref Expression
mrcss ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → (𝐹𝑈) ⊆ (𝐹𝑉))

Proof of Theorem mrcss
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 sstr2 4001 . . . . . 6 (𝑈𝑉 → (𝑉𝑠𝑈𝑠))
21adantr 480 . . . . 5 ((𝑈𝑉𝑠𝐶) → (𝑉𝑠𝑈𝑠))
32ss2rabdv 4085 . . . 4 (𝑈𝑉 → {𝑠𝐶𝑉𝑠} ⊆ {𝑠𝐶𝑈𝑠})
4 intss 4973 . . . 4 ({𝑠𝐶𝑉𝑠} ⊆ {𝑠𝐶𝑈𝑠} → {𝑠𝐶𝑈𝑠} ⊆ {𝑠𝐶𝑉𝑠})
53, 4syl 17 . . 3 (𝑈𝑉 {𝑠𝐶𝑈𝑠} ⊆ {𝑠𝐶𝑉𝑠})
653ad2ant2 1133 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → {𝑠𝐶𝑈𝑠} ⊆ {𝑠𝐶𝑉𝑠})
7 simp1 1135 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → 𝐶 ∈ (Moore‘𝑋))
8 sstr 4003 . . . 4 ((𝑈𝑉𝑉𝑋) → 𝑈𝑋)
983adant1 1129 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → 𝑈𝑋)
10 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
1110mrcval 17654 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
127, 9, 11syl2anc 584 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
1310mrcval 17654 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉𝑋) → (𝐹𝑉) = {𝑠𝐶𝑉𝑠})
14133adant2 1130 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → (𝐹𝑉) = {𝑠𝐶𝑉𝑠})
156, 12, 143sstr4d 4042 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → (𝐹𝑈) ⊆ (𝐹𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1536  wcel 2105  {crab 3432  wss 3962   cint 4950  cfv 6562  Moorecmre 17626  mrClscmrc 17627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-mre 17630  df-mrc 17631
This theorem is referenced by:  mrcsscl  17664  mrcuni  17665  mrcssd  17668  ismrc  42688  isnacs3  42697
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