Theorem ntrclsfveq1 41670

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
ntrclsfveq1	⊢ (𝜑 → ((𝐼‘𝑆) = 𝐶 ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ 𝐶)))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝑗,𝐾,𝑘   𝑆,𝑗   𝜑,𝑖,𝑗,𝑘

Allowed substitution hints:   𝐶(𝑖,𝑗,𝑘)   𝐷(𝑖,𝑗,𝑘)   𝑆(𝑖,𝑘)   𝐼(𝑖,𝑗,𝑘)   𝐾(𝑖)   𝑂(𝑖,𝑗,𝑘)

Proof of Theorem ntrclsfveq1

Step	Hyp	Ref	Expression
1		ntrclsfv.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵)
2	1	elpwid 4544	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
3		dfss4 4192	. . . . 5 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐶)) = 𝐶)
4	2, 3	sylib 217	. . . 4 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ 𝐶)) = 𝐶)
5	4	eqcomd 2744	. . 3 ⊢ (𝜑 → 𝐶 = (𝐵 ∖ (𝐵 ∖ 𝐶)))
6	5	eqeq2d 2749	. 2 ⊢ (𝜑 → ((𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) = 𝐶 ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ (𝐵 ∖ 𝐶))))
7		ntrcls.o	. . . 4 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
8		ntrcls.d	. . . 4 ⊢ 𝐷 = (𝑂‘𝐵)
9		ntrcls.r	. . . 4 ⊢ (𝜑 → 𝐼𝐷𝐾)
10		ntrclsfv.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
11	7, 8, 9, 10	ntrclsfv 41669	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))
12	11	eqeq1d 2740	. 2 ⊢ (𝜑 → ((𝐼‘𝑆) = 𝐶 ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) = 𝐶))
13	7, 8, 9	ntrclskex 41664	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
14		elmapi 8637	. . . . . 6 ⊢ (𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
15	13, 14	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
16	8, 9	ntrclsrcomplex 41645	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
17	15, 16	ffvelrnd 6962	. . . 4 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑆)) ∈ 𝒫 𝐵)
18	17	elpwid 4544	. . 3 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵)
19		difssd 4067	. . 3 ⊢ (𝜑 → (𝐵 ∖ 𝐶) ⊆ 𝐵)
20		rcompleq 4229	. . 3 ⊢ (((𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵 ∧ (𝐵 ∖ 𝐶) ⊆ 𝐵) → ((𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ 𝐶) ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ (𝐵 ∖ 𝐶))))
21	18, 19, 20	syl2anc 584	. 2 ⊢ (𝜑 → ((𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ 𝐶) ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ (𝐵 ∖ 𝐶))))
22	6, 12, 21	3bitr4d 311	1 ⊢ (𝜑 → ((𝐼‘𝑆) = 𝐶 ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887  𝒫 cpw 4533   class class class wbr 5074   ↦ cmpt 5157  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617

This theorem is referenced by:  ntrclsfveq  41672