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Theorem ntrclsss 44076
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 44072 . . 3 (𝜑 → (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))))
6 ntrclsfv.t . . . 4 (𝜑𝑇 ∈ 𝒫 𝐵)
71, 2, 3, 6ntrclsfv 44072 . . 3 (𝜑 → (𝐼𝑇) = (𝐵 ∖ (𝐾‘(𝐵𝑇))))
85, 7sseq12d 4017 . 2 (𝜑 → ((𝐼𝑆) ⊆ (𝐼𝑇) ↔ (𝐵 ∖ (𝐾‘(𝐵𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵𝑇)))))
91, 2, 3ntrclskex 44067 . . . 4 (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵))
109ancli 548 . . 3 (𝜑 → (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)))
11 elmapi 8889 . . . . . . 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 44048 . . . . . . 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 4609 . . . 4 ((𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (𝐾‘(𝐵𝑇)) ⊆ 𝐵)
172, 3ntrclsrcomplex 44048 . . . . . . 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 4609 . . . 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 4304 . . 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 1540  wcel 2108  Vcvv 3480  cdif 3948  wss 3951  𝒫 cpw 4600   class class class wbr 5143  cmpt 5225  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868
This theorem is referenced by: (None)
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