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Theorem ntrclsfveq 38862
Description: If interior and closure functions are related then equality of a pair of function values is equivalent to equality 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
ntrclsfveq (𝜑 → ((𝐼𝑆) = (𝐼𝑇) ↔ (𝐾‘(𝐵𝑆)) = (𝐾‘(𝐵𝑇))))
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 ntrclsfveq
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.t . . . 4 (𝜑𝑇 ∈ 𝒫 𝐵)
51, 2, 3, 4ntrclsfv 38859 . . 3 (𝜑 → (𝐼𝑇) = (𝐵 ∖ (𝐾‘(𝐵𝑇))))
65eqeq2d 2770 . 2 (𝜑 → ((𝐼𝑆) = (𝐼𝑇) ↔ (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑇)))))
7 ntrclsfv.s . . 3 (𝜑𝑆 ∈ 𝒫 𝐵)
82, 3ntrclsrcomplex 38835 . . 3 (𝜑 → (𝐵 ∖ (𝐾‘(𝐵𝑇))) ∈ 𝒫 𝐵)
91, 2, 3, 7, 8ntrclsfveq1 38860 . 2 (𝜑 → ((𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑇))) ↔ (𝐾‘(𝐵𝑆)) = (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵𝑇))))))
101, 2, 3ntrclskex 38854 . . . . . . 7 (𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
11 elmapi 8045 . . . . . . 7 (𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
1210, 11syl 17 . . . . . 6 (𝜑𝐾:𝒫 𝐵⟶𝒫 𝐵)
132, 3ntrclsrcomplex 38835 . . . . . 6 (𝜑 → (𝐵𝑇) ∈ 𝒫 𝐵)
1412, 13ffvelrnd 6523 . . . . 5 (𝜑 → (𝐾‘(𝐵𝑇)) ∈ 𝒫 𝐵)
1514elpwid 4314 . . . 4 (𝜑 → (𝐾‘(𝐵𝑇)) ⊆ 𝐵)
16 dfss4 4001 . . . 4 ((𝐾‘(𝐵𝑇)) ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵𝑇)))) = (𝐾‘(𝐵𝑇)))
1715, 16sylib 208 . . 3 (𝜑 → (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵𝑇)))) = (𝐾‘(𝐵𝑇)))
1817eqeq2d 2770 . 2 (𝜑 → ((𝐾‘(𝐵𝑆)) = (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵𝑇)))) ↔ (𝐾‘(𝐵𝑆)) = (𝐾‘(𝐵𝑇))))
196, 9, 183bitrd 294 1 (𝜑 → ((𝐼𝑆) = (𝐼𝑇) ↔ (𝐾‘(𝐵𝑆)) = (𝐾‘(𝐵𝑇))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = 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 6813  𝑚 cmap 8023
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 7114
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 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025
This theorem is referenced by: (None)
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