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Theorem ntrclsss 44053
Description: If interior and closure functions are related then a subset relation of a pair of function values is equivalent to subset relation of a pair of the other function's values. (Contributed by RP, 27-Jun-2021.)
Hypotheses
Ref Expression
ntrcls.o 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
ntrcls.d 𝐷 = (𝑂𝐵)
ntrcls.r (𝜑𝐼𝐷𝐾)
ntrclsfv.s (𝜑𝑆 ∈ 𝒫 𝐵)
ntrclsfv.t (𝜑𝑇 ∈ 𝒫 𝐵)
Assertion
Ref Expression
ntrclsss (𝜑 → ((𝐼𝑆) ⊆ (𝐼𝑇) ↔ (𝐾‘(𝐵𝑇)) ⊆ (𝐾‘(𝐵𝑆))))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝑗,𝐾,𝑘   𝑆,𝑗   𝑇,𝑗   𝜑,𝑖,𝑗,𝑘
Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘)   𝑆(𝑖,𝑘)   𝑇(𝑖,𝑘)   𝐼(𝑖,𝑗,𝑘)   𝐾(𝑖)   𝑂(𝑖,𝑗,𝑘)

Proof of Theorem ntrclsss
StepHypRef Expression
1 ntrcls.o . . . 4 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
2 ntrcls.d . . . 4 𝐷 = (𝑂𝐵)
3 ntrcls.r . . . 4 (𝜑𝐼𝐷𝐾)
4 ntrclsfv.s . . . 4 (𝜑𝑆 ∈ 𝒫 𝐵)
51, 2, 3, 4ntrclsfv 44049 . . 3 (𝜑 → (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))))
6 ntrclsfv.t . . . 4 (𝜑𝑇 ∈ 𝒫 𝐵)
71, 2, 3, 6ntrclsfv 44049 . . 3 (𝜑 → (𝐼𝑇) = (𝐵 ∖ (𝐾‘(𝐵𝑇))))
85, 7sseq12d 4029 . 2 (𝜑 → ((𝐼𝑆) ⊆ (𝐼𝑇) ↔ (𝐵 ∖ (𝐾‘(𝐵𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵𝑇)))))
91, 2, 3ntrclskex 44044 . . . 4 (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵))
109ancli 548 . . 3 (𝜑 → (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)))
11 elmapi 8888 . . . . . . 7 (𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
1211adantl 481 . . . . . 6 ((𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
132, 3ntrclsrcomplex 44025 . . . . . . 7 (𝜑 → (𝐵𝑇) ∈ 𝒫 𝐵)
1413adantr 480 . . . . . 6 ((𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (𝐵𝑇) ∈ 𝒫 𝐵)
1512, 14ffvelcdmd 7105 . . . . 5 ((𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (𝐾‘(𝐵𝑇)) ∈ 𝒫 𝐵)
1615elpwid 4614 . . . 4 ((𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (𝐾‘(𝐵𝑇)) ⊆ 𝐵)
172, 3ntrclsrcomplex 44025 . . . . . . 7 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
1817adantr 480 . . . . . 6 ((𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (𝐵𝑆) ∈ 𝒫 𝐵)
1912, 18ffvelcdmd 7105 . . . . 5 ((𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (𝐾‘(𝐵𝑆)) ∈ 𝒫 𝐵)
2019elpwid 4614 . . . 4 ((𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (𝐾‘(𝐵𝑆)) ⊆ 𝐵)
2116, 20jca 511 . . 3 ((𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)) → ((𝐾‘(𝐵𝑇)) ⊆ 𝐵 ∧ (𝐾‘(𝐵𝑆)) ⊆ 𝐵))
22 sscon34b 4310 . . 3 (((𝐾‘(𝐵𝑇)) ⊆ 𝐵 ∧ (𝐾‘(𝐵𝑆)) ⊆ 𝐵) → ((𝐾‘(𝐵𝑇)) ⊆ (𝐾‘(𝐵𝑆)) ↔ (𝐵 ∖ (𝐾‘(𝐵𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵𝑇)))))
2310, 21, 223syl 18 . 2 (𝜑 → ((𝐾‘(𝐵𝑇)) ⊆ (𝐾‘(𝐵𝑆)) ↔ (𝐵 ∖ (𝐾‘(𝐵𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵𝑇)))))
248, 23bitr4d 282 1 (𝜑 → ((𝐼𝑆) ⊆ (𝐼𝑇) ↔ (𝐾‘(𝐵𝑇)) ⊆ (𝐾‘(𝐵𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cdif 3960  wss 3963  𝒫 cpw 4605   class class class wbr 5148  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867
This theorem is referenced by: (None)
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