Theorem ntrclsfv1 42796

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is a functional relation between them (Contributed by RP, 28-May-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)

Assertion

Ref	Expression
ntrclsfv1	⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)

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

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

Proof of Theorem ntrclsfv1

Step	Hyp	Ref	Expression
1		ntrcls.r	. 2 ⊢ (𝜑 → 𝐼𝐷𝐾)
2		ntrcls.o	. . . . . . 7 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
3		ntrcls.d	. . . . . . 7 ⊢ 𝐷 = (𝑂‘𝐵)
4	2, 3, 1	ntrclsf1o 42792	. . . . . 6 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
5		f1ofn 6834	. . . . . 6 ⊢ (𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵))
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → 𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵))
7	2, 3, 1	ntrclsiex 42794	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
8	6, 7	jca 512	. . . 4 ⊢ (𝜑 → (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)))
9		fnfun 6649	. . . . . 6 ⊢ (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → Fun 𝐷)
10	9	adantr 481	. . . . 5 ⊢ ((𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → Fun 𝐷)
11		fndm 6652	. . . . . . 7 ⊢ (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → dom 𝐷 = (𝒫 𝐵 ↑m 𝒫 𝐵))
12	11	eleq2d 2819	. . . . . 6 ⊢ (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → (𝐼 ∈ dom 𝐷 ↔ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)))
13	12	biimpar 478	. . . . 5 ⊢ ((𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → 𝐼 ∈ dom 𝐷)
14	10, 13	jca 512	. . . 4 ⊢ ((𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (Fun 𝐷 ∧ 𝐼 ∈ dom 𝐷))
15	8, 14	syl 17	. . 3 ⊢ (𝜑 → (Fun 𝐷 ∧ 𝐼 ∈ dom 𝐷))
16		funbrfvb 6946	. . 3 ⊢ ((Fun 𝐷 ∧ 𝐼 ∈ dom 𝐷) → ((𝐷‘𝐼) = 𝐾 ↔ 𝐼𝐷𝐾))
17	15, 16	syl 17	. 2 ⊢ (𝜑 → ((𝐷‘𝐼) = 𝐾 ↔ 𝐼𝐷𝐾))
18	1, 17	mpbird 256	1 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∖ cdif 3945  𝒫 cpw 4602   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  Fun wfun 6537   Fn wfn 6538  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7408   ↑_m cmap 8819

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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-map 8821

This theorem is referenced by:  ntrclsfv2  42797  ntrclscls00  42807  ntrclsiso  42808  ntrclsk2  42809  ntrclskb  42810  ntrclsk3  42811  ntrclsk13  42812