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Theorem ntrclsss 37381
 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 37377 . . 3 (𝜑 → (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))))
6 ntrclsfv.t . . . 4 (𝜑𝑇 ∈ 𝒫 𝐵)
71, 2, 3, 6ntrclsfv 37377 . . 3 (𝜑 → (𝐼𝑇) = (𝐵 ∖ (𝐾‘(𝐵𝑇))))
85, 7sseq12d 3597 . 2 (𝜑 → ((𝐼𝑆) ⊆ (𝐼𝑇) ↔ (𝐵 ∖ (𝐾‘(𝐵𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵𝑇)))))
91, 2, 3ntrclskex 37372 . . . 4 (𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
109ancli 572 . . 3 (𝜑 → (𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)))
11 elmapi 7765 . . . . . . 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 37353 . . . . . . 7 (𝜑 → (𝐵𝑇) ∈ 𝒫 𝐵)
1413adantr 480 . . . . . 6 ((𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐵𝑇) ∈ 𝒫 𝐵)
1512, 14ffvelrnd 6268 . . . . 5 ((𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐾‘(𝐵𝑇)) ∈ 𝒫 𝐵)
1615elpwid 4118 . . . 4 ((𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐾‘(𝐵𝑇)) ⊆ 𝐵)
172, 3ntrclsrcomplex 37353 . . . . . . 7 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
1817adantr 480 . . . . . 6 ((𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐵𝑆) ∈ 𝒫 𝐵)
1912, 18ffvelrnd 6268 . . . . 5 ((𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐾‘(𝐵𝑆)) ∈ 𝒫 𝐵)
2019elpwid 4118 . . . 4 ((𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐾‘(𝐵𝑆)) ⊆ 𝐵)
2116, 20jca 553 . . 3 ((𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → ((𝐾‘(𝐵𝑇)) ⊆ 𝐵 ∧ (𝐾‘(𝐵𝑆)) ⊆ 𝐵))
22 sscon34b 37337 . . 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 270 1 (𝜑 → ((𝐼𝑆) ⊆ (𝐼𝑇) ↔ (𝐾‘(𝐵𝑇)) ⊆ (𝐾‘(𝐵𝑆))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746 This theorem is referenced by: (None)
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