Theorem ntrclsss 44056

Description: If interior and closure functions are related then a subset relation of a pair of function values is equivalent to subset relation 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
ntrclsss	⊢ (𝜑 → ((𝐼‘𝑆) ⊆ (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆))))

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 ntrclsss

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.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
5	1, 2, 3, 4	ntrclsfv 44052	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))
6		ntrclsfv.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵)
7	1, 2, 3, 6	ntrclsfv 44052	. . 3 ⊢ (𝜑 → (𝐼‘𝑇) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))))
8	5, 7	sseq12d 3969	. 2 ⊢ (𝜑 → ((𝐼‘𝑆) ⊆ (𝐼‘𝑇) ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))))
9	1, 2, 3	ntrclskex 44047	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
10	9	ancli 548	. . 3 ⊢ (𝜑 → (𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)))
11		elmapi 8776	. . . . . . 7 ⊢ (𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
12	11	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
13	2, 3	ntrclsrcomplex 44028	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∖ 𝑇) ∈ 𝒫 𝐵)
14	13	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐵 ∖ 𝑇) ∈ 𝒫 𝐵)
15	12, 14	ffvelcdmd 7019	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑇)) ∈ 𝒫 𝐵)
16	15	elpwid 4560	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵)
17	2, 3	ntrclsrcomplex 44028	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
18	17	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
19	12, 18	ffvelcdmd 7019	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑆)) ∈ 𝒫 𝐵)
20	19	elpwid 4560	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵)
21	16, 20	jca 511	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → ((𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵 ∧ (𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵))
22		sscon34b 4255	. . 3 ⊢ (((𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵 ∧ (𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵) → ((𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆)) ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))))
23	10, 21, 22	3syl 18	. 2 ⊢ (𝜑 → ((𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆)) ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))))
24	8, 23	bitr4d 282	1 ⊢ (𝜑 → ((𝐼‘𝑆) ⊆ (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436   ∖ cdif 3900   ⊆ wss 3903  𝒫 cpw 4551   class class class wbr 5092   ↦ cmpt 5173  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ↑_m cmap 8753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-map 8755

This theorem is referenced by: (None)