Theorem ntrclsfveq1 44042

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 4568	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
3		dfss4 4228	. . . . 5 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐶)) = 𝐶)
4	2, 3	sylib 218	. . . 4 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ 𝐶)) = 𝐶)
5	4	eqcomd 2735	. . 3 ⊢ (𝜑 → 𝐶 = (𝐵 ∖ (𝐵 ∖ 𝐶)))
6	5	eqeq2d 2740	. 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 44041	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))
12	11	eqeq1d 2731	. 2 ⊢ (𝜑 → ((𝐼‘𝑆) = 𝐶 ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) = 𝐶))
13	7, 8, 9	ntrclskex 44036	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
14		elmapi 8799	. . . . . 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 44017	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
17	15, 16	ffvelcdmd 7039	. . . 4 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑆)) ∈ 𝒫 𝐵)
18	17	elpwid 4568	. . 3 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵)
19		difssd 4096	. . 3 ⊢ (𝜑 → (𝐵 ∖ 𝐶) ⊆ 𝐵)
20		rcompleq 4264	. . 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 206   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ∖ cdif 3908   ⊆ wss 3911  𝒫 cpw 4559   class class class wbr 5102   ↦ cmpt 5183  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ↑_m cmap 8776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-map 8778

This theorem is referenced by:  ntrclsfveq  44044