MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mrcss Structured version   Visualization version   GIF version

Theorem mrcss 16323
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 3643 . . . . . 6 (𝑈𝑉 → (𝑉𝑠𝑈𝑠))
21adantr 480 . . . . 5 ((𝑈𝑉𝑠𝐶) → (𝑉𝑠𝑈𝑠))
32ss2rabdv 3716 . . . 4 (𝑈𝑉 → {𝑠𝐶𝑉𝑠} ⊆ {𝑠𝐶𝑈𝑠})
4 intss 4530 . . . 4 ({𝑠𝐶𝑉𝑠} ⊆ {𝑠𝐶𝑈𝑠} → {𝑠𝐶𝑈𝑠} ⊆ {𝑠𝐶𝑉𝑠})
53, 4syl 17 . . 3 (𝑈𝑉 {𝑠𝐶𝑈𝑠} ⊆ {𝑠𝐶𝑉𝑠})
653ad2ant2 1103 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → {𝑠𝐶𝑈𝑠} ⊆ {𝑠𝐶𝑉𝑠})
7 simp1 1081 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → 𝐶 ∈ (Moore‘𝑋))
8 sstr 3644 . . . 4 ((𝑈𝑉𝑉𝑋) → 𝑈𝑋)
983adant1 1099 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → 𝑈𝑋)
10 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
1110mrcval 16317 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
127, 9, 11syl2anc 694 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
1310mrcval 16317 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉𝑋) → (𝐹𝑉) = {𝑠𝐶𝑉𝑠})
14133adant2 1100 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → (𝐹𝑉) = {𝑠𝐶𝑉𝑠})
156, 12, 143sstr4d 3681 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → (𝐹𝑈) ⊆ (𝐹𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1054   = wceq 1523  wcel 2030  {crab 2945  wss 3607   cint 4507  cfv 5926  Moorecmre 16289  mrClscmrc 16290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-mre 16293  df-mrc 16294
This theorem is referenced by:  mrcsscl  16327  mrcuni  16328  mrcssd  16331  ismrc  37581  isnacs3  37590
  Copyright terms: Public domain W3C validator