Theorem ntrclsfveq 44052

Description: If interior and closure functions are related then equality of a pair of function values is equivalent to equality of a pair of the other function's values. (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.t	⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵)

Assertion

Ref	Expression
ntrclsfveq	⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇))))

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

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

Proof of Theorem ntrclsfveq

Step	Hyp	Ref	Expression
1		ntrcls.o	. . . 4 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
2		ntrcls.d	. . . 4 ⊢ 𝐷 = (𝑂‘𝐵)
3		ntrcls.r	. . . 4 ⊢ (𝜑 → 𝐼𝐷𝐾)
4		ntrclsfv.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵)
5	1, 2, 3, 4	ntrclsfv 44049	. . 3 ⊢ (𝜑 → (𝐼‘𝑇) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))))
6	5	eqeq2d 2746	. 2 ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))))
7		ntrclsfv.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
8	2, 3	ntrclsrcomplex 44025	. . 3 ⊢ (𝜑 → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))) ∈ 𝒫 𝐵)
9	1, 2, 3, 7, 8	ntrclsfveq1 44050	. 2 ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))))))
10	1, 2, 3	ntrclskex 44044	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
11		elmapi 8888	. . . . . . 7 ⊢ (𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
12	10, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
13	2, 3	ntrclsrcomplex 44025	. . . . . 6 ⊢ (𝜑 → (𝐵 ∖ 𝑇) ∈ 𝒫 𝐵)
14	12, 13	ffvelcdmd 7105	. . . . 5 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑇)) ∈ 𝒫 𝐵)
15	14	elpwid 4614	. . . 4 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵)
16		dfss4 4275	. . . 4 ⊢ ((𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) = (𝐾‘(𝐵 ∖ 𝑇)))
17	15, 16	sylib 218	. . 3 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) = (𝐾‘(𝐵 ∖ 𝑇)))
18	17	eqeq2d 2746	. 2 ⊢ (𝜑 → ((𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇))))
19	6, 9, 18	3bitrd 305	1 ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇))))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ∖ cdif 3960   ⊆ wss 3963  𝒫 cpw 4605   class class class wbr 5148   ↦ cmpt 5231  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ↑_m cmap 8865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867

This theorem is referenced by: (None)