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Theorem ntrclsss 38881
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 38877 . . 3 (𝜑 → (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))))
6 ntrclsfv.t . . . 4 (𝜑𝑇 ∈ 𝒫 𝐵)
71, 2, 3, 6ntrclsfv 38877 . . 3 (𝜑 → (𝐼𝑇) = (𝐵 ∖ (𝐾‘(𝐵𝑇))))
85, 7sseq12d 3775 . 2 (𝜑 → ((𝐼𝑆) ⊆ (𝐼𝑇) ↔ (𝐵 ∖ (𝐾‘(𝐵𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵𝑇)))))
91, 2, 3ntrclskex 38872 . . . 4 (𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
109ancli 575 . . 3 (𝜑 → (𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)))
11 elmapi 8047 . . . . . . 7 (𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
1211adantl 473 . . . . . 6 ((𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
132, 3ntrclsrcomplex 38853 . . . . . . 7 (𝜑 → (𝐵𝑇) ∈ 𝒫 𝐵)
1413adantr 472 . . . . . 6 ((𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐵𝑇) ∈ 𝒫 𝐵)
1512, 14ffvelrnd 6524 . . . . 5 ((𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐾‘(𝐵𝑇)) ∈ 𝒫 𝐵)
1615elpwid 4314 . . . 4 ((𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐾‘(𝐵𝑇)) ⊆ 𝐵)
172, 3ntrclsrcomplex 38853 . . . . . . 7 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
1817adantr 472 . . . . . 6 ((𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐵𝑆) ∈ 𝒫 𝐵)
1912, 18ffvelrnd 6524 . . . . 5 ((𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐾‘(𝐵𝑆)) ∈ 𝒫 𝐵)
2019elpwid 4314 . . . 4 ((𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (𝐾‘(𝐵𝑆)) ⊆ 𝐵)
2116, 20jca 555 . . 3 ((𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → ((𝐾‘(𝐵𝑇)) ⊆ 𝐵 ∧ (𝐾‘(𝐵𝑆)) ⊆ 𝐵))
22 sscon34b 38837 . . 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 271 1 (𝜑 → ((𝐼𝑆) ⊆ (𝐼𝑇) ↔ (𝐾‘(𝐵𝑇)) ⊆ (𝐾‘(𝐵𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cdif 3712  wss 3715  𝒫 cpw 4302   class class class wbr 4804  cmpt 4881  wf 6045  cfv 6049  (class class class)co 6814  𝑚 cmap 8025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027
This theorem is referenced by: (None)
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