Theorem ntrclsfv1 43382

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 43378	. . . . . 6 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
5		f1ofn 6828	. . . . . 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 43380	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
8	6, 7	jca 511	. . . 4 ⊢ (𝜑 → (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)))
9		fnfun 6643	. . . . . 6 ⊢ (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → Fun 𝐷)
10	9	adantr 480	. . . . 5 ⊢ ((𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → Fun 𝐷)
11		fndm 6646	. . . . . . 7 ⊢ (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → dom 𝐷 = (𝒫 𝐵 ↑m 𝒫 𝐵))
12	11	eleq2d 2813	. . . . . 6 ⊢ (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → (𝐼 ∈ dom 𝐷 ↔ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)))
13	12	biimpar 477	. . . . 5 ⊢ ((𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → 𝐼 ∈ dom 𝐷)
14	10, 13	jca 511	. . . 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 6940	. . 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 257	1 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468   ∖ cdif 3940  𝒫 cpw 4597   class class class wbr 5141   ↦ cmpt 5224  dom cdm 5669  Fun wfun 6531   Fn wfn 6532  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7405   ↑_m cmap 8822

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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-map 8824

This theorem is referenced by:  ntrclsfv2  43383  ntrclscls00  43393  ntrclsiso  43394  ntrclsk2  43395  ntrclskb  43396  ntrclsk3  43397  ntrclsk13  43398