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Theorem ntrclsfveq 42746
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 42743 . . 3 (𝜑 → (𝐼𝑇) = (𝐵 ∖ (𝐾‘(𝐵𝑇))))
65eqeq2d 2744 . 2 (𝜑 → ((𝐼𝑆) = (𝐼𝑇) ↔ (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑇)))))
7 ntrclsfv.s . . 3 (𝜑𝑆 ∈ 𝒫 𝐵)
82, 3ntrclsrcomplex 42719 . . 3 (𝜑 → (𝐵 ∖ (𝐾‘(𝐵𝑇))) ∈ 𝒫 𝐵)
91, 2, 3, 7, 8ntrclsfveq1 42744 . 2 (𝜑 → ((𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑇))) ↔ (𝐾‘(𝐵𝑆)) = (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵𝑇))))))
101, 2, 3ntrclskex 42738 . . . . . . 7 (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵))
11 elmapi 8839 . . . . . . 7 (𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
1210, 11syl 17 . . . . . 6 (𝜑𝐾:𝒫 𝐵⟶𝒫 𝐵)
132, 3ntrclsrcomplex 42719 . . . . . 6 (𝜑 → (𝐵𝑇) ∈ 𝒫 𝐵)
1412, 13ffvelcdmd 7083 . . . . 5 (𝜑 → (𝐾‘(𝐵𝑇)) ∈ 𝒫 𝐵)
1514elpwid 4610 . . . 4 (𝜑 → (𝐾‘(𝐵𝑇)) ⊆ 𝐵)
16 dfss4 4257 . . . 4 ((𝐾‘(𝐵𝑇)) ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵𝑇)))) = (𝐾‘(𝐵𝑇)))
1715, 16sylib 217 . . 3 (𝜑 → (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵𝑇)))) = (𝐾‘(𝐵𝑇)))
1817eqeq2d 2744 . 2 (𝜑 → ((𝐾‘(𝐵𝑆)) = (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵𝑇)))) ↔ (𝐾‘(𝐵𝑆)) = (𝐾‘(𝐵𝑇))))
196, 9, 183bitrd 305 1 (𝜑 → ((𝐼𝑆) = (𝐼𝑇) ↔ (𝐾‘(𝐵𝑆)) = (𝐾‘(𝐵𝑇))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  Vcvv 3475  cdif 3944  wss 3947  𝒫 cpw 4601   class class class wbr 5147  cmpt 5230  wf 6536  cfv 6540  (class class class)co 7404  m cmap 8816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7970  df-2nd 7971  df-map 8818
This theorem is referenced by: (None)
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