Theorem ntrclsfveq1 42646

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 4606	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
3		dfss4 4255	. . . . 5 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐶)) = 𝐶)
4	2, 3	sylib 217	. . . 4 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ 𝐶)) = 𝐶)
5	4	eqcomd 2738	. . 3 ⊢ (𝜑 → 𝐶 = (𝐵 ∖ (𝐵 ∖ 𝐶)))
6	5	eqeq2d 2743	. 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 42645	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))
12	11	eqeq1d 2734	. 2 ⊢ (𝜑 → ((𝐼‘𝑆) = 𝐶 ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) = 𝐶))
13	7, 8, 9	ntrclskex 42640	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
14		elmapi 8828	. . . . . 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 42621	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
17	15, 16	ffvelcdmd 7073	. . . 4 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑆)) ∈ 𝒫 𝐵)
18	17	elpwid 4606	. . 3 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵)
19		difssd 4129	. . 3 ⊢ (𝜑 → (𝐵 ∖ 𝐶) ⊆ 𝐵)
20		rcompleq 4292	. . 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 310	1 ⊢ (𝜑 → ((𝐼‘𝑆) = 𝐶 ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∖ cdif 3942   ⊆ wss 3945  𝒫 cpw 4597   class class class wbr 5142   ↦ cmpt 5225  ⟶wf 6529  ‘cfv 6533  (class class class)co 7394   ↑_m cmap 8805

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 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7959  df-2nd 7960  df-map 8807

This theorem is referenced by:  ntrclsfveq  42648