Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ntrclsfveq2 Structured version   Visualization version   GIF version

Theorem ntrclsfveq2 42315
Description: If interior and closure functions are related then specific function values are complementary. (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.c (𝜑𝐶 ∈ 𝒫 𝐵)
Assertion
Ref Expression
ntrclsfveq2 (𝜑 → ((𝐼‘(𝐵𝑆)) = 𝐶 ↔ (𝐾𝑆) = (𝐵𝐶)))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝑗,𝐼,𝑘   𝑆,𝑗   𝜑,𝑖,𝑗,𝑘
Allowed substitution hints:   𝐶(𝑖,𝑗,𝑘)   𝐷(𝑖,𝑗,𝑘)   𝑆(𝑖,𝑘)   𝐼(𝑖)   𝐾(𝑖,𝑗,𝑘)   𝑂(𝑖,𝑗,𝑘)

Proof of Theorem ntrclsfveq2
StepHypRef Expression
1 ntrcls.o . . . . . . 7 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
2 ntrcls.d . . . . . . 7 𝐷 = (𝑂𝐵)
3 ntrcls.r . . . . . . 7 (𝜑𝐼𝐷𝐾)
41, 2, 3ntrclsiex 42307 . . . . . 6 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
5 elmapi 8786 . . . . . 6 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
64, 5syl 17 . . . . 5 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
72, 3ntrclsrcomplex 42289 . . . . 5 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
86, 7ffvelcdmd 7035 . . . 4 (𝜑 → (𝐼‘(𝐵𝑆)) ∈ 𝒫 𝐵)
98elpwid 4569 . . 3 (𝜑 → (𝐼‘(𝐵𝑆)) ⊆ 𝐵)
10 ntrclsfv.c . . . 4 (𝜑𝐶 ∈ 𝒫 𝐵)
1110elpwid 4569 . . 3 (𝜑𝐶𝐵)
12 rcompleq 4255 . . 3 (((𝐼‘(𝐵𝑆)) ⊆ 𝐵𝐶𝐵) → ((𝐼‘(𝐵𝑆)) = 𝐶 ↔ (𝐵 ∖ (𝐼‘(𝐵𝑆))) = (𝐵𝐶)))
139, 11, 12syl2anc 584 . 2 (𝜑 → ((𝐼‘(𝐵𝑆)) = 𝐶 ↔ (𝐵 ∖ (𝐼‘(𝐵𝑆))) = (𝐵𝐶)))
141, 2, 3ntrclsnvobr 42306 . . . 4 (𝜑𝐾𝐷𝐼)
15 ntrclsfv.s . . . 4 (𝜑𝑆 ∈ 𝒫 𝐵)
161, 2, 14, 15ntrclsfv 42313 . . 3 (𝜑 → (𝐾𝑆) = (𝐵 ∖ (𝐼‘(𝐵𝑆))))
1716eqeq1d 2738 . 2 (𝜑 → ((𝐾𝑆) = (𝐵𝐶) ↔ (𝐵 ∖ (𝐼‘(𝐵𝑆))) = (𝐵𝐶)))
1813, 17bitr4d 281 1 (𝜑 → ((𝐼‘(𝐵𝑆)) = 𝐶 ↔ (𝐾𝑆) = (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  Vcvv 3445  cdif 3907  wss 3910  𝒫 cpw 4560   class class class wbr 5105  cmpt 5188  wf 6492  cfv 6496  (class class class)co 7356  m cmap 8764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7920  df-2nd 7921  df-map 8766
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator