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Theorem ntrclsss 40398
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 40394 . . 3 (𝜑 → (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))))
6 ntrclsfv.t . . . 4 (𝜑𝑇 ∈ 𝒫 𝐵)
71, 2, 3, 6ntrclsfv 40394 . . 3 (𝜑 → (𝐼𝑇) = (𝐵 ∖ (𝐾‘(𝐵𝑇))))
85, 7sseq12d 3998 . 2 (𝜑 → ((𝐼𝑆) ⊆ (𝐼𝑇) ↔ (𝐵 ∖ (𝐾‘(𝐵𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵𝑇)))))
91, 2, 3ntrclskex 40389 . . . 4 (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵))
109ancli 551 . . 3 (𝜑 → (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)))
11 elmapi 8420 . . . . . . 7 (𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
1211adantl 484 . . . . . 6 ((𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
132, 3ntrclsrcomplex 40370 . . . . . . 7 (𝜑 → (𝐵𝑇) ∈ 𝒫 𝐵)
1413adantr 483 . . . . . 6 ((𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (𝐵𝑇) ∈ 𝒫 𝐵)
1512, 14ffvelrnd 6845 . . . . 5 ((𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (𝐾‘(𝐵𝑇)) ∈ 𝒫 𝐵)
1615elpwid 4551 . . . 4 ((𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (𝐾‘(𝐵𝑇)) ⊆ 𝐵)
172, 3ntrclsrcomplex 40370 . . . . . . 7 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
1817adantr 483 . . . . . 6 ((𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (𝐵𝑆) ∈ 𝒫 𝐵)
1912, 18ffvelrnd 6845 . . . . 5 ((𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (𝐾‘(𝐵𝑆)) ∈ 𝒫 𝐵)
2019elpwid 4551 . . . 4 ((𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (𝐾‘(𝐵𝑆)) ⊆ 𝐵)
2116, 20jca 514 . . 3 ((𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵)) → ((𝐾‘(𝐵𝑇)) ⊆ 𝐵 ∧ (𝐾‘(𝐵𝑆)) ⊆ 𝐵))
22 sscon34b 40354 . . 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 284 1 (𝜑 → ((𝐼𝑆) ⊆ (𝐼𝑇) ↔ (𝐾‘(𝐵𝑇)) ⊆ (𝐾‘(𝐵𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  Vcvv 3493  cdif 3931  wss 3934  𝒫 cpw 4537   class class class wbr 5057  cmpt 5137  wf 6344  cfv 6348  (class class class)co 7148  m cmap 8398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  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 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  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-reu 3143  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-iun 4912  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-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-map 8400
This theorem is referenced by: (None)
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