Theorem ntrclsfveq2 44302

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 44294	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
5		elmapi 8786	. . . . . 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 44276	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
8	6, 7	ffvelcdmd 7030	. . . 4 ⊢ (𝜑 → (𝐼‘(𝐵 ∖ 𝑆)) ∈ 𝒫 𝐵)
9	8	elpwid 4563	. . 3 ⊢ (𝜑 → (𝐼‘(𝐵 ∖ 𝑆)) ⊆ 𝐵)
10		ntrclsfv.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵)
11	10	elpwid 4563	. . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
12		rcompleq 4257	. . 3 ⊢ (((𝐼‘(𝐵 ∖ 𝑆)) ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵) → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ 𝐶)))
13	9, 11, 12	syl2anc 584	. 2 ⊢ (𝜑 → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ 𝐶)))
14	1, 2, 3	ntrclsnvobr 44293	. . . 4 ⊢ (𝜑 → 𝐾𝐷𝐼)
15		ntrclsfv.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
16	1, 2, 14, 15	ntrclsfv 44300	. . 3 ⊢ (𝜑 → (𝐾‘𝑆) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆))))
17	16	eqeq1d 2738	. 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 1541   ∈ wcel 2113  Vcvv 3440   ∖ cdif 3898   ⊆ wss 3901  𝒫 cpw 4554   class class class wbr 5098   ↦ cmpt 5179  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765

This theorem is referenced by: (None)