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Theorem ntrclsfveq 43302
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 ↦ (π‘˜ ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 βˆ– (π‘˜β€˜(𝑖 βˆ– 𝑗))))))
ntrcls.d 𝐷 = (π‘‚β€˜π΅)
ntrcls.r (πœ‘ β†’ 𝐼𝐷𝐾)
ntrclsfv.s (πœ‘ β†’ 𝑆 ∈ 𝒫 𝐡)
ntrclsfv.t (πœ‘ β†’ 𝑇 ∈ 𝒫 𝐡)
Assertion
Ref Expression
ntrclsfveq (πœ‘ β†’ ((πΌβ€˜π‘†) = (πΌβ€˜π‘‡) ↔ (πΎβ€˜(𝐡 βˆ– 𝑆)) = (πΎβ€˜(𝐡 βˆ– 𝑇))))
Distinct variable groups:   𝐡,𝑖,𝑗,π‘˜   𝑗,𝐾,π‘˜   𝑆,𝑗   𝑇,𝑗   πœ‘,𝑖,𝑗,π‘˜
Allowed substitution hints:   𝐷(𝑖,𝑗,π‘˜)   𝑆(𝑖,π‘˜)   𝑇(𝑖,π‘˜)   𝐼(𝑖,𝑗,π‘˜)   𝐾(𝑖)   𝑂(𝑖,𝑗,π‘˜)

Proof of Theorem ntrclsfveq
StepHypRef Expression
1 ntrcls.o . . . 4 𝑂 = (𝑖 ∈ V ↦ (π‘˜ ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 βˆ– (π‘˜β€˜(𝑖 βˆ– 𝑗))))))
2 ntrcls.d . . . 4 𝐷 = (π‘‚β€˜π΅)
3 ntrcls.r . . . 4 (πœ‘ β†’ 𝐼𝐷𝐾)
4 ntrclsfv.t . . . 4 (πœ‘ β†’ 𝑇 ∈ 𝒫 𝐡)
51, 2, 3, 4ntrclsfv 43299 . . 3 (πœ‘ β†’ (πΌβ€˜π‘‡) = (𝐡 βˆ– (πΎβ€˜(𝐡 βˆ– 𝑇))))
65eqeq2d 2735 . 2 (πœ‘ β†’ ((πΌβ€˜π‘†) = (πΌβ€˜π‘‡) ↔ (πΌβ€˜π‘†) = (𝐡 βˆ– (πΎβ€˜(𝐡 βˆ– 𝑇)))))
7 ntrclsfv.s . . 3 (πœ‘ β†’ 𝑆 ∈ 𝒫 𝐡)
82, 3ntrclsrcomplex 43275 . . 3 (πœ‘ β†’ (𝐡 βˆ– (πΎβ€˜(𝐡 βˆ– 𝑇))) ∈ 𝒫 𝐡)
91, 2, 3, 7, 8ntrclsfveq1 43300 . 2 (πœ‘ β†’ ((πΌβ€˜π‘†) = (𝐡 βˆ– (πΎβ€˜(𝐡 βˆ– 𝑇))) ↔ (πΎβ€˜(𝐡 βˆ– 𝑆)) = (𝐡 βˆ– (𝐡 βˆ– (πΎβ€˜(𝐡 βˆ– 𝑇))))))
101, 2, 3ntrclskex 43294 . . . . . . 7 (πœ‘ β†’ 𝐾 ∈ (𝒫 𝐡 ↑m 𝒫 𝐡))
11 elmapi 8839 . . . . . . 7 (𝐾 ∈ (𝒫 𝐡 ↑m 𝒫 𝐡) β†’ 𝐾:𝒫 π΅βŸΆπ’« 𝐡)
1210, 11syl 17 . . . . . 6 (πœ‘ β†’ 𝐾:𝒫 π΅βŸΆπ’« 𝐡)
132, 3ntrclsrcomplex 43275 . . . . . 6 (πœ‘ β†’ (𝐡 βˆ– 𝑇) ∈ 𝒫 𝐡)
1412, 13ffvelcdmd 7077 . . . . 5 (πœ‘ β†’ (πΎβ€˜(𝐡 βˆ– 𝑇)) ∈ 𝒫 𝐡)
1514elpwid 4603 . . . 4 (πœ‘ β†’ (πΎβ€˜(𝐡 βˆ– 𝑇)) βŠ† 𝐡)
16 dfss4 4250 . . . 4 ((πΎβ€˜(𝐡 βˆ– 𝑇)) βŠ† 𝐡 ↔ (𝐡 βˆ– (𝐡 βˆ– (πΎβ€˜(𝐡 βˆ– 𝑇)))) = (πΎβ€˜(𝐡 βˆ– 𝑇)))
1715, 16sylib 217 . . 3 (πœ‘ β†’ (𝐡 βˆ– (𝐡 βˆ– (πΎβ€˜(𝐡 βˆ– 𝑇)))) = (πΎβ€˜(𝐡 βˆ– 𝑇)))
1817eqeq2d 2735 . 2 (πœ‘ β†’ ((πΎβ€˜(𝐡 βˆ– 𝑆)) = (𝐡 βˆ– (𝐡 βˆ– (πΎβ€˜(𝐡 βˆ– 𝑇)))) ↔ (πΎβ€˜(𝐡 βˆ– 𝑆)) = (πΎβ€˜(𝐡 βˆ– 𝑇))))
196, 9, 183bitrd 305 1 (πœ‘ β†’ ((πΌβ€˜π‘†) = (πΌβ€˜π‘‡) ↔ (πΎβ€˜(𝐡 βˆ– 𝑆)) = (πΎβ€˜(𝐡 βˆ– 𝑇))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  Vcvv 3466   βˆ– cdif 3937   βŠ† wss 3940  π’« cpw 4594   class class class wbr 5138   ↦ cmpt 5221  βŸΆwf 6529  β€˜cfv 6533  (class class class)co 7401   ↑m cmap 8816
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-map 8818
This theorem is referenced by: (None)
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