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Theorem ntrclsfveq 38886
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 38883 . . 3 (𝜑 → (𝐼𝑇) = (𝐵 ∖ (𝐾‘(𝐵𝑇))))
65eqeq2d 2781 . 2 (𝜑 → ((𝐼𝑆) = (𝐼𝑇) ↔ (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑇)))))
7 ntrclsfv.s . . 3 (𝜑𝑆 ∈ 𝒫 𝐵)
82, 3ntrclsrcomplex 38859 . . 3 (𝜑 → (𝐵 ∖ (𝐾‘(𝐵𝑇))) ∈ 𝒫 𝐵)
91, 2, 3, 7, 8ntrclsfveq1 38884 . 2 (𝜑 → ((𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑇))) ↔ (𝐾‘(𝐵𝑆)) = (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵𝑇))))))
101, 2, 3ntrclskex 38878 . . . . . . 7 (𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
11 elmapi 8031 . . . . . . 7 (𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
1210, 11syl 17 . . . . . 6 (𝜑𝐾:𝒫 𝐵⟶𝒫 𝐵)
132, 3ntrclsrcomplex 38859 . . . . . 6 (𝜑 → (𝐵𝑇) ∈ 𝒫 𝐵)
1412, 13ffvelrnd 6503 . . . . 5 (𝜑 → (𝐾‘(𝐵𝑇)) ∈ 𝒫 𝐵)
1514elpwid 4309 . . . 4 (𝜑 → (𝐾‘(𝐵𝑇)) ⊆ 𝐵)
16 dfss4 4007 . . . 4 ((𝐾‘(𝐵𝑇)) ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵𝑇)))) = (𝐾‘(𝐵𝑇)))
1715, 16sylib 208 . . 3 (𝜑 → (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵𝑇)))) = (𝐾‘(𝐵𝑇)))
1817eqeq2d 2781 . 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 1631  wcel 2145  Vcvv 3351  cdif 3720  wss 3723  𝒫 cpw 4297   class class class wbr 4786  cmpt 4863  wf 6027  cfv 6031  (class class class)co 6793  𝑚 cmap 8009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-map 8011
This theorem is referenced by: (None)
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