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Theorem ntrclsfveq 40280
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 40277 . . 3 (𝜑 → (𝐼𝑇) = (𝐵 ∖ (𝐾‘(𝐵𝑇))))
65eqeq2d 2837 . 2 (𝜑 → ((𝐼𝑆) = (𝐼𝑇) ↔ (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑇)))))
7 ntrclsfv.s . . 3 (𝜑𝑆 ∈ 𝒫 𝐵)
82, 3ntrclsrcomplex 40253 . . 3 (𝜑 → (𝐵 ∖ (𝐾‘(𝐵𝑇))) ∈ 𝒫 𝐵)
91, 2, 3, 7, 8ntrclsfveq1 40278 . 2 (𝜑 → ((𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑇))) ↔ (𝐾‘(𝐵𝑆)) = (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵𝑇))))))
101, 2, 3ntrclskex 40272 . . . . . . 7 (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵))
11 elmapi 8418 . . . . . . 7 (𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
1210, 11syl 17 . . . . . 6 (𝜑𝐾:𝒫 𝐵⟶𝒫 𝐵)
132, 3ntrclsrcomplex 40253 . . . . . 6 (𝜑 → (𝐵𝑇) ∈ 𝒫 𝐵)
1412, 13ffvelrnd 6848 . . . . 5 (𝜑 → (𝐾‘(𝐵𝑇)) ∈ 𝒫 𝐵)
1514elpwid 4556 . . . 4 (𝜑 → (𝐾‘(𝐵𝑇)) ⊆ 𝐵)
16 dfss4 4239 . . . 4 ((𝐾‘(𝐵𝑇)) ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵𝑇)))) = (𝐾‘(𝐵𝑇)))
1715, 16sylib 219 . . 3 (𝜑 → (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵𝑇)))) = (𝐾‘(𝐵𝑇)))
1817eqeq2d 2837 . 2 (𝜑 → ((𝐾‘(𝐵𝑆)) = (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵𝑇)))) ↔ (𝐾‘(𝐵𝑆)) = (𝐾‘(𝐵𝑇))))
196, 9, 183bitrd 306 1 (𝜑 → ((𝐼𝑆) = (𝐼𝑇) ↔ (𝐾‘(𝐵𝑆)) = (𝐾‘(𝐵𝑇))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  wcel 2107  Vcvv 3500  cdif 3937  wss 3940  𝒫 cpw 4542   class class class wbr 5063  cmpt 5143  wf 6348  cfv 6352  (class class class)co 7148  m cmap 8396
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 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-map 8398
This theorem is referenced by: (None)
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