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Theorem mrcss 16879
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 3972 . . . . . 6 (𝑈𝑉 → (𝑉𝑠𝑈𝑠))
21adantr 483 . . . . 5 ((𝑈𝑉𝑠𝐶) → (𝑉𝑠𝑈𝑠))
32ss2rabdv 4050 . . . 4 (𝑈𝑉 → {𝑠𝐶𝑉𝑠} ⊆ {𝑠𝐶𝑈𝑠})
4 intss 4888 . . . 4 ({𝑠𝐶𝑉𝑠} ⊆ {𝑠𝐶𝑈𝑠} → {𝑠𝐶𝑈𝑠} ⊆ {𝑠𝐶𝑉𝑠})
53, 4syl 17 . . 3 (𝑈𝑉 {𝑠𝐶𝑈𝑠} ⊆ {𝑠𝐶𝑉𝑠})
653ad2ant2 1129 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → {𝑠𝐶𝑈𝑠} ⊆ {𝑠𝐶𝑉𝑠})
7 simp1 1131 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → 𝐶 ∈ (Moore‘𝑋))
8 sstr 3973 . . . 4 ((𝑈𝑉𝑉𝑋) → 𝑈𝑋)
983adant1 1125 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → 𝑈𝑋)
10 mrcfval.f . . . 4 𝐹 = (mrCls‘𝐶)
1110mrcval 16873 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
127, 9, 11syl2anc 586 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → (𝐹𝑈) = {𝑠𝐶𝑈𝑠})
1310mrcval 16873 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑉𝑋) → (𝐹𝑉) = {𝑠𝐶𝑉𝑠})
14133adant2 1126 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → (𝐹𝑉) = {𝑠𝐶𝑉𝑠})
156, 12, 143sstr4d 4012 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈𝑉𝑉𝑋) → (𝐹𝑈) ⊆ (𝐹𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1082   = wceq 1531  wcel 2108  {crab 3140  wss 3934   cint 4867  cfv 6348  Moorecmre 16845  mrClscmrc 16846
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-mre 16849  df-mrc 16850
This theorem is referenced by:  mrcsscl  16883  mrcuni  16884  mrcssd  16887  ismrc  39289  isnacs3  39298
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