Theorem ntrclsfveq2 44023

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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
ntrcls.d	⊢ 𝐷 = (𝑂‘𝐵)
ntrcls.r	⊢ (𝜑 → 𝐼𝐷𝐾)
ntrclsfv.s	⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
ntrclsfv.c	⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵)

Assertion

Ref	Expression
ntrclsfveq2	⊢ (𝜑 → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐾‘𝑆) = (𝐵 ∖ 𝐶)))

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

Step	Hyp	Ref	Expression
1		ntrcls.o	. . . . . . 7 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
2		ntrcls.d	. . . . . . 7 ⊢ 𝐷 = (𝑂‘𝐵)
3		ntrcls.r	. . . . . . 7 ⊢ (𝜑 → 𝐼𝐷𝐾)
4	1, 2, 3	ntrclsiex 44015	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
5		elmapi 8907	. . . . . 6 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
7	2, 3	ntrclsrcomplex 43997	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
8	6, 7	ffvelcdmd 7119	. . . 4 ⊢ (𝜑 → (𝐼‘(𝐵 ∖ 𝑆)) ∈ 𝒫 𝐵)
9	8	elpwid 4631	. . 3 ⊢ (𝜑 → (𝐼‘(𝐵 ∖ 𝑆)) ⊆ 𝐵)
10		ntrclsfv.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵)
11	10	elpwid 4631	. . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
12		rcompleq 4324	. . 3 ⊢ (((𝐼‘(𝐵 ∖ 𝑆)) ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵) → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ 𝐶)))
13	9, 11, 12	syl2anc 583	. 2 ⊢ (𝜑 → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ 𝐶)))
14	1, 2, 3	ntrclsnvobr 44014	. . . 4 ⊢ (𝜑 → 𝐾𝐷𝐼)
15		ntrclsfv.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
16	1, 2, 14, 15	ntrclsfv 44021	. . 3 ⊢ (𝜑 → (𝐾‘𝑆) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆))))
17	16	eqeq1d 2742	. 2 ⊢ (𝜑 → ((𝐾‘𝑆) = (𝐵 ∖ 𝐶) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ 𝐶)))
18	13, 17	bitr4d 282	1 ⊢ (𝜑 → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐾‘𝑆) = (𝐵 ∖ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  𝒫 cpw 4622   class class class wbr 5166   ↦ cmpt 5249  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-map 8886

This theorem is referenced by: (None)