Theorem ntrclsfveq2 43301

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 43293	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
5		elmapi 8839	. . . . . 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 43275	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
8	6, 7	ffvelcdmd 7077	. . . 4 ⊢ (𝜑 → (𝐼‘(𝐵 ∖ 𝑆)) ∈ 𝒫 𝐵)
9	8	elpwid 4603	. . 3 ⊢ (𝜑 → (𝐼‘(𝐵 ∖ 𝑆)) ⊆ 𝐵)
10		ntrclsfv.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵)
11	10	elpwid 4603	. . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
12		rcompleq 4287	. . 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 43292	. . . 4 ⊢ (𝜑 → 𝐾𝐷𝐼)
15		ntrclsfv.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
16	1, 2, 14, 15	ntrclsfv 43299	. . 3 ⊢ (𝜑 → (𝐾‘𝑆) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆))))
17	16	eqeq1d 2726	. 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 205   = wceq 1533   ∈ wcel 2098  Vcvv 3466   ∖ cdif 3937   ⊆ wss 3940  𝒫 cpw 4594   class class class wbr 5138   ↦ cmpt 5221  ⟶wf 6529  ‘cfv 6533  (class class class)co 7401   ↑_m cmap 8816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  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 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-map 8818

This theorem is referenced by: (None)