Theorem ntrclsss 44644

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 44640	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))
6		ntrclsfv.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵)
7	1, 2, 3, 6	ntrclsfv 44640	. . 3 ⊢ (𝜑 → (𝐼‘𝑇) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))))
8	5, 7	sseq12d 3970	. 2 ⊢ (𝜑 → ((𝐼‘𝑆) ⊆ (𝐼‘𝑇) ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ⊆ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))))
9	1, 2, 3	ntrclskex 44635	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
10	9	ancli 556	. . 3 ⊢ (𝜑 → (𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)))
11		elmapi 8830	. . . . . . 7 ⊢ (𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
12	11	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
13	2, 3	ntrclsrcomplex 44616	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∖ 𝑇) ∈ 𝒫 𝐵)
14	13	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐵 ∖ 𝑇) ∈ 𝒫 𝐵)
15	12, 14	ffvelcdmd 7066	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑇)) ∈ 𝒫 𝐵)
16	15	elpwid 4565	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵)
17	2, 3	ntrclsrcomplex 44616	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
18	17	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
19	12, 18	ffvelcdmd 7066	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑆)) ∈ 𝒫 𝐵)
20	19	elpwid 4565	. . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵)
21	16, 20	jca 519	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → ((𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵 ∧ (𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵))
22		sscon34b 4257	. . 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 284	1 ⊢ (𝜑 → ((𝐼‘𝑆) ⊆ (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑇)) ⊆ (𝐾‘(𝐵 ∖ 𝑆))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1561   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905  𝒫 cpw 4556   class class class wbr 5101   ↦ cmpt 5182  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396   ↑_m cmap 8808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-map 8810

This theorem is referenced by: (None)